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Incomplete gamma function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the upper and lower incomplete gamma functions are types of <a href="Special_functions" title="Special functions">special functions</a> which arise as solutions to various mathematical problems such as certain <a href="Integral" title="Integral">integrals</a>.

Their respective names stem from their integral definitions, which are defined similarly to the <a href="Gamma_function" title="Gamma function">gamma function</a> but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The upper incomplete gamma function is defined as:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)=\int _{x}^{\infty }t^{s-1}\,e^{-t}\,dt,}</annotation>

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whereas the lower incomplete gamma function is defined as:

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)=\int _{0}^{x}t^{s-1}\,e^{-t}\,dt.}</annotation>

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In both cases s is a complex parameter, such that the real part of s is positive.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

By <a href="Integration_by_parts" title="Integration by parts">integration by parts</a> we find the <a href="Recurrence_relations" class="mw-redirect" title="Recurrence relations">recurrence relations</a>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s+1,x)=s\Gamma (s,x)+x^{s}e^{-x}}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s+1,x)=s\gamma (s,x)-x^{s}e^{-x}.}</annotation>

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Since the ordinary gamma function is defined as

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\int _{0}^{\infty }t^{s-1}\,e^{-t}\,dt}</annotation>

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we have

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\Gamma (s,0)=\lim _{x\to \infty }\gamma (s,x)}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)+\Gamma (s,x)=\Gamma (s).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Continuation_to_complex_values">Continuation to complex values</h3></div>

The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a>, with respect both to x and s, defined for almost all combinations of complex x and s.<a href="#cite_note-auto3-1">[1]</a> Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.

<div class="mw-heading mw-heading4"><h4 id="Lower_incomplete_gamma_function">Lower incomplete gamma function</h4></div>

<div class="mw-heading mw-heading5"><h5 id="Holomorphic_extension">Holomorphic extension</h5></div>

Repeated application of the recurrence relation for the lower incomplete gamma function leads to the <a href="Power_series" title="Power series">power series</a> expansion: <a href="#cite_note-auto2-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)=\sum _{k=0}^{\infty }{\frac {x^{s}e^{-x}x^{k}}{s(s+1)\cdots (s+k)}}=x^{s}\,\Gamma (s)\,e^{-x}\sum _{k=0}^{\infty }{\frac {x^{k}}{\Gamma (s+k+1)}}.}</annotation>

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Given the rapid growth in <a href="Absolute_value#Complex_numbers" title="Absolute value">absolute value</a> of Γ(z + k) when k → ∞, and the fact that the <a href="Reciprocal_Gamma_function" class="mw-redirect" title="Reciprocal Gamma function">reciprocal of Γ(z)</a> is an <a href="Entire_function" title="Entire function">entire function</a>, the coefficients in the rightmost sum are well-defined, and locally the sum <a href="Uniform_convergence" title="Uniform convergence">converges uniformly</a> for all complex s and x. By a theorem of <a href="Weierstrass" class="mw-redirect" title="Weierstrass">Weierstrass</a>,<a href="#cite_note-class_notes-3">[3]</a> the limiting function, sometimes denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation>

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</math><img src="./b94d9c54dd2c77e4571ed51bd635158b22c204a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}" loading="lazy">,<a href="#cite_note-auto1-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}(s,z):=e^{-z}\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (s+k+1)}}}</annotation>

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is <a href="Entire_function" title="Entire function">entire</a> with respect to both z (for fixed s) and s (for fixed z),<a href="#cite_note-auto3-1">[1]</a> and, thus, holomorphic on C × C by <a href="Hartogs's_theorem_on_separate_holomorphicity" title="Hartogs's theorem on separate holomorphicity">Hartogs' theorem</a>.<a href="#cite_note-5">[5]</a> Hence, the following decomposition<a href="#cite_note-auto3-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)=z^{s}\,\Gamma (s)\,\gamma ^{*}(s,z),}</annotation>

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extends the real lower incomplete gamma function as a <a href="Holomorphic_function" title="Holomorphic function">holomorphic function</a>, both jointly and separately in z and s. It follows from the properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{s}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{s}}</annotation>

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</math><img src="./c53db4564e89c001721430a555f619f0e9c9ac54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.094ex; height:2.343ex;" alt="{\displaystyle z^{s}}" loading="lazy"> and the <a href="Gamma_function" title="Gamma function">Γ-function</a>, that the first two factors capture the <a href="Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (s,z)}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)}</annotation>

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</math><img src="./d4b9af5c0733be3f13ca9d7e52b04877b2876199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.284ex; height:2.843ex;" alt="{\displaystyle \gamma (s,z)}" loading="lazy"> (at z = 0 or s a non-positive integer), whereas the last factor contributes to its zeros.

<div class="mw-heading mw-heading5"><h5 id="Multi-valuedness">Multi-valuedness</h5></div>

The <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> log z = log |z| + i arg z is determined up to a multiple of 2πi only, which renders it <a href="Multi-valued_function" class="mw-redirect" title="Multi-valued function">multi-valued</a>. Functions involving the complex logarithm typically inherit this property. Among these are the <a href="Exponentiation#nth_roots_of_a_complex_number" title="Exponentiation">complex power</a>, and, since zs appears in its decomposition, the γ-function, too.

The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Strategies to handle this are:

<ul><li>(the most general way) replace the domain C of multi-valued functions by a suitable manifold in C × C called <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>. While this removes multi-valuedness, one has to know the theory behind it;<a href="#cite_note-6">[6]</a></li>

<li>restrict the domain such that a multi-valued function decomposes into separate single-valued <a href="Branch_point" title="Branch point">branches</a>, which can be handled individually.</li></ul>

The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:

<div class="mw-heading mw-heading6"><h6 id="Sectors">Sectors</h6></div>

Sectors in C having their vertex at z = 0 often prove to be appropriate domains for complex expressions. A sector D consists of all complex z fulfilling z ≠ 0 and α − δ < arg z < α + δ with some α and 0 < δ ≤ π. Often, α can be arbitrarily chosen and is not specified then. If δ is not given, it is assumed to be π, and the sector is in fact the whole plane C, with the exception of a half-line originating at z = 0 and pointing into the direction of −α, usually serving as a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>. Note: In many applications and texts, α is silently taken to be 0, which centers the sector around the positive real axis.

<div class="mw-heading mw-heading6"><h6 id="Branches">Branches</h6></div>

In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range (α − δ, α + δ). Based on such a restricted logarithm, zs and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on D (or C×D), called branches of their multi-valued counterparts on D. Adding a multiple of 2π to α yields a different set of correlated branches on the same set D. However, in any given context here, α is assumed fixed and all branches involved are associated to it. If |α| < δ, the branches are called <a href="Principal_branch" title="Principal branch">principal</a>, because they equal their real analogues on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.

<div class="mw-heading mw-heading6"><h6 id="Relation_between_branches">Relation between branches</h6></div>

The values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2\pi iks}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{2\pi iks}}</annotation>

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</math><img src="./e2c518ea1f934a3cc7a48d50d9bd96ac4e3b3678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.275ex; height:2.676ex;" alt="{\displaystyle e^{2\pi iks}}" loading="lazy">,<a href="#cite_note-auto3-1">[1]</a> for k a suitable integer.

<div class="mw-heading mw-heading5"><h5 id="Behavior_near_branch_point">Behavior near branch point</h5></div>

The decomposition above further shows, that γ behaves near z = 0 <a href="Asymptotic" class="mw-redirect" title="Asymptotic">asymptotically</a> like:

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)\asymp z^{s}\,\Gamma (s)\,\gamma ^{*}(s,0)=z^{s}\,\Gamma (s)/\Gamma (s+1)=z^{s}/s.}</annotation>

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For positive real x, y and s, xy/y → 0, when (x, y) → (0, s). This seems to justify setting γ(s, 0) = 0 for real s > 0. However, matters are somewhat different in the complex realm. Only if (a) the real part of s is positive, and (b) values uv are taken from just a <a href="Finite_set" title="Finite set">finite set</a> of branches, they are guaranteed to converge to zero as (u, v) → (0, s), and so does γ(u, v). On a single <a href="Branch_point" title="Branch point">branch</a> of γ(b) is naturally fulfilled, so there γ(s, 0) = 0 for s with positive real part is a <a href="Continuous_function" title="Continuous function">continuous limit</a>. Also note that such a continuation is by no means an <a href="Analytic_continuation" title="Analytic continuation">analytic one</a>.

<div class="mw-heading mw-heading5"><h5 id="Algebraic_relations">Algebraic relations</h5></div>

All algebraic relations and differential equations observed by the real γ(s, z) hold for its holomorphic counterpart as well. This is a consequence of the <a href="Identity_theorem" title="Identity theorem">identity theorem</a>, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation <a href="#cite_note-auto2-2">[2]</a> and ∂γ(s, z)/∂z = zs−1 e−z <a href="#cite_note-auto2-2">[2]</a> are preserved on corresponding branches.

<div class="mw-heading mw-heading5"><h5 id="Integral_representation">Integral representation</h5></div>

The last relation tells us, that, for fixed s, γ is a <a href="Primitive_function" class="mw-redirect" title="Primitive function">primitive or antiderivative</a> of the holomorphic function zs−1 e−z. Consequently, for any complex u, v ≠ 0,

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<annotation encoding="application/x-tex">{\displaystyle \int _{u}^{v}t^{s-1}\,e^{-t}\,dt=\gamma (s,v)-\gamma (s,u)}</annotation>

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holds, as long as the <a href="Line_integral" title="Line integral">path of integration</a> is entirely contained in the domain of a branch of the integrand. If, additionally, the real part of s is positive, then the limit γ(s, u) → 0 for u → 0 applies, finally arriving at the complex integral definition of γ<a href="#cite_note-auto3-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)=\int _{0}^{z}t^{s-1}\,e^{-t}\,dt,\,\Re (s)>0.}</annotation>

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Any path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting 0 and z.

<div class="mw-heading mw-heading5"><h5 id="Limit_for_z_→_+∞">Limit for z → +∞</h5></div>

<div class="mw-heading mw-heading6"><h6 id="Real_values">Real values</h6></div>

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:<a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\int _{0}^{\infty }t^{s-1}\,e^{-t}\,dt=\lim _{x\to \infty }\gamma (s,x)}</annotation>

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<div class="mw-heading mw-heading6"><h6 id="s_complex">s complex</h6></div>

This result extends to complex s. Assume first 1 ≤ Re(s) ≤ 2 and 1 < a < b. Then

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<annotation encoding="application/x-tex">{\displaystyle \left|\gamma (s,b)-\gamma (s,a)\right|\leq \int _{a}^{b}\left|t^{s-1}\right|e^{-t}\,dt=\int _{a}^{b}t^{\Re s-1}e^{-t}\,dt\leq \int _{a}^{b}te^{-t}\,dt}</annotation>

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where<a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle \left|z^{s}\right|=\left|z\right|^{\Re s}\,e^{-\Im s\arg z}}</annotation>

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has been used in the middle. Since the final integral becomes arbitrarily small if only a is large enough, γ(s, x) converges uniformly for x → ∞ on the strip 1 ≤ Re(s) ≤ 2 towards a holomorphic function,<a href="#cite_note-class_notes-3">[3]</a> which must be Γ(s) because of the identity theorem. Taking the limit in the recurrence relation γ(s, x) = (s − 1) γ(s − 1, x) − xs − 1 e−x and noting, that lim xn e−x = 0 for x → ∞ and all n, shows, that γ(s, x) converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\lim _{x\to \infty }\gamma (s,x)}</annotation>

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for all complex s not a non-positive integer, x real and γ principal.

<div class="mw-heading mw-heading6"><h6 id="Sectorwise_convergence">Sectorwise convergence</h6></div>

Now let u be from the sector |arg z| < δ < π/2 with some fixed δ (α = 0), γ be the principal branch on this sector, and look at

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)-\gamma (s,u)=\Gamma (s)-\gamma (s,|u|)+\gamma (s,|u|)-\gamma (s,u).}</annotation>

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As shown above, the first difference can be made arbitrarily small, if |u| is sufficiently large. The second difference allows for following estimation:

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<annotation encoding="application/x-tex">{\displaystyle \left|\gamma (s,|u|)-\gamma (s,u)\right|\leq \int _{u}^{|u|}\left|z^{s-1}e^{-z}\right|dz=\int _{u}^{|u|}\left|z\right|^{\Re s-1}\,e^{-\Im s\,\arg z}\,e^{-\Re z}\,dz,}</annotation>

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where we made use of the integral representation of γ and the formula about |zs| above. If we integrate along the arc with radius R = |u| around 0 connecting u and |u|, then the last integral is

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<annotation encoding="application/x-tex">{\displaystyle \leq R\left|\arg u\right|R^{\Re s-1}\,e^{\Im s\,|\arg u|}\,e^{-R\cos \arg u}\leq \delta \,R^{\Re s}\,e^{\Im s\,\delta }\,e^{-R\cos \delta }=M\,(R\,\cos \delta )^{\Re s}\,e^{-R\cos \delta }}</annotation>

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where M = δ(cos δ)−Re s eIm sδ is a constant independent of u or R. Again referring to the behavior of xn e−x for large x, we see that the last expression approaches 0 as R increases towards ∞.

In total we now have:

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)=\lim _{|z|\to \infty }\gamma (s,z),\quad \left|\arg z\right|<\pi /2-\epsilon ,}</annotation>

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if s is not a non-negative integer, 0 < ε < π/2 is arbitrarily small, but fixed, and γ denotes the principal branch on this domain.

<div class="mw-heading mw-heading5"><h5 id="Overview">Overview</h5></div>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)}</annotation>

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</math><img src="./d4b9af5c0733be3f13ca9d7e52b04877b2876199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.284ex; height:2.843ex;" alt="{\displaystyle \gamma (s,z)}" loading="lazy"> is:

<ul><li><a href="Entire_function" title="Entire function">entire</a> in z for fixed, positive integer s;</li>

<li>multi-valued <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> in z for fixed s not an integer, with a <a href="Branch_point" title="Branch point">branch point</a> at z = 0;</li>

<li>on each branch <a href="Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> in s for fixed z ≠ 0, with simple poles at non-positive integers s.</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Upper_incomplete_gamma_function">Upper incomplete gamma function</h4></div>

As for the upper incomplete gamma function, a <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> extension, with respect to z or s, is given by<a href="#cite_note-auto3-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)=\Gamma (s)-\gamma (s,z)}</annotation>

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at points (s, z), where the right hand side exists. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is multi-valued, the same holds for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">, but a restriction to principal values only yields the single-valued principal branch of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy">.

When s is a non-positive integer in the above equation, neither part of the difference is defined, and a <a href="Limit_of_a_function" title="Limit of a function">limiting process</a>, here developed for s → 0, fills in the missing values. <a href="Complex_analysis" title="Complex analysis">Complex analysis</a> guarantees <a href="Holomorphic_function" title="Holomorphic function">holomorphicity</a>, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)}</annotation>

</semantics>

</math><img src="./73d13f4a2398613d31737fc8e16c293d4b58e3ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.475ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,z)}" loading="lazy"> proves to be <a href="Bounded_function" title="Bounded function">bounded</a> in a <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of that limit for a fixed z.

To determine the limit, the power series of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation>

</semantics>

</math><img src="./b94d9c54dd2c77e4571ed51bd635158b22c204a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}" loading="lazy"> at z = 0 is useful. When replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-x}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-x}}</annotation>

</semantics>

</math><img src="./b201e900a30da19a1f1e4bdddcc70fe7e502be4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.535ex; height:2.509ex;" alt="{\displaystyle e^{-x}}" loading="lazy"> by its power series in the integral definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">, one obtains (assume x,s positive reals for now):

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)=\int _{0}^{x}t^{s-1}e^{-t}\,dt=\int _{0}^{x}\sum _{k=0}^{\infty }\left(-1\right)^{k}\,{\frac {t^{s+k-1}}{k!}}\,dt=\sum _{k=0}^{\infty }\left(-1\right)^{k}\,{\frac {x^{s+k}}{k!(s+k)}}=x^{s}\,\sum _{k=0}^{\infty }{\frac {(-x)^{k}}{k!(s+k)}}}</annotation>

</semantics>

</math>

or<a href="#cite_note-auto1-4">[4]</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}(s,x)=\sum _{k=0}^{\infty }{\frac {(-x)^{k}}{k!\,\Gamma (s)(s+k)}},}</annotation>

</semantics>

</math>

which, as a series representation of the entire <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation>

</semantics>

</math><img src="./b94d9c54dd2c77e4571ed51bd635158b22c204a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}" loading="lazy"> function, converges for all complex x (and all complex s not a non-positive integer).

With its restriction to real values lifted, the series allows the expansion:

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)-{\frac {1}{s}}=-{\frac {1}{s}}+z^{s}\,\sum _{k=0}^{\infty }{\frac {(-z)^{k}}{k!(s+k)}}={\frac {z^{s}-1}{s}}+z^{s}\,\sum _{k=1}^{\infty }{\frac {\left(-z\right)^{k}}{k!(s+k)}},\quad \Re (s)>-1,\,s\neq 0.}</annotation>

</semantics>

</math>

When s → 0:<a href="#cite_note-9">[9]</a>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{s}-1}{s}}\to \ln(z),\quad \Gamma (s)-{\frac {1}{s}}={\frac {1}{s}}-\gamma +O(s)-{\frac {1}{s}}\to -\gamma ,}</annotation>

</semantics>

</math>

(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is the <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> here), hence,

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (0,z)=\lim _{s\to 0}\left(\Gamma (s)-{\tfrac {1}{s}}-(\gamma (s,z)-{\tfrac {1}{s}})\right)=-\gamma -\ln(z)-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\,(k!)}}}</annotation>

</semantics>

</math>

is the limiting function to the upper incomplete gamma function as s → 0, also known as the <a href="Exponential_integral" title="Exponential integral">exponential integral</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}(z)}</annotation>

</semantics>

</math><img src="./99a8a5cda07ddc0773ff859981fe68f9212a49ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.667ex; height:2.843ex;" alt="{\displaystyle E_{1}(z)}" loading="lazy">.<a href="#cite_note-10">[10]</a>

By way of the recurrence relation, values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (-n,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (-n,z)}</annotation>

</semantics>

</math><img src="./df8f021e0dbaa009c585dc9d378b45cd4311535f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.587ex; height:2.843ex;" alt="{\displaystyle \Gamma (-n,z)}" loading="lazy"> for positive integers n can be derived from this result,<a href="#cite_note-11">[11]</a>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (-n,z)={\frac {1}{n!}}\left({\frac {e^{-z}}{z^{n}}}\sum _{k=0}^{n-1}(-1)^{k}(n-k-1)!\,z^{k}+\left(-1\right)^{n}\Gamma (0,z)\right)}</annotation>

</semantics>

</math>

so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to z and s, for all s and z ≠ 0.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)}</annotation>

</semantics>

</math><img src="./73d13f4a2398613d31737fc8e16c293d4b58e3ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.475ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,z)}" loading="lazy"> is:

<ul><li><a href="Entire_function" title="Entire function">entire</a> in z for fixed, positive integral s;</li>

<li>multi-valued <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> in z for fixed s non zero and not a positive integer, with a <a href="Branch_point" title="Branch point">branch point</a> at z = 0;</li>

<li>equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s)}</annotation>

</semantics>

</math><img src="./7b5e07eea9dfb68e7d8417e60d6d7768982061fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.353ex; height:2.843ex;" alt="{\displaystyle \Gamma (s)}" loading="lazy"> for s with positive real part and z = 0 (the limit when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s_{i},z_{i})\to (s,0)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s_{i},z_{i})\to (s,0)}</annotation>

</semantics>

</math><img src="./6e081c74573f6e695e9ead5a0654e85074ca7644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.324ex; height:2.843ex;" alt="{\displaystyle (s_{i},z_{i})\to (s,0)}" loading="lazy">), but this is a continuous extension, not an <a href="Analytic_continuation" title="Analytic continuation">analytic one</a> (does not hold for real s < 0!);</li>

<li>on each branch <a href="Entire_function" title="Entire function">entire</a> in s for fixed z ≠ 0.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Special_values">Special values</h3></div>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s+1,1)={\frac {\lfloor es!\rfloor }{e}}}</annotation>

</semantics>

</math><img src="./e7913b6db50e50216fe40828472bfcda4110e9dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.372ex; height:5.676ex;" alt="{\displaystyle \Gamma (s+1,1)={\frac {\lfloor es!\rfloor }{e}}}" loading="lazy"> if s is a positive <a href="Integer" title="Integer">integer</a>,</li>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)=(s-1)!\,e^{-x}\sum _{k=0}^{s-1}{\frac {x^{k}}{k!}}}</annotation>

</semantics>

</math><img src="./9c991677495fffc15cbe94294c8ff4c32935028f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.669ex; height:7.509ex;" alt="{\displaystyle \Gamma (s,x)=(s-1)!\,e^{-x}\sum _{k=0}^{s-1}{\frac {x^{k}}{k!}}}" loading="lazy"> if s is a positive <a href="Integer" title="Integer">integer</a>,<a href="#cite_note-12">[12]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,0)=\Gamma (s),\Re (s)>0}</annotation>

</semantics>

</math><img src="./7a293e0661f7e5ba551956ae64bc15564f517567.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.119ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,0)=\Gamma (s),\Re (s)>0}" loading="lazy">,</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (1,x)=e^{-x}}</annotation>

</semantics>

</math><img src="./1e78fe3721b5dd4ebcf19941474d53f6b4f208b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.421ex; height:3.009ex;" alt="{\displaystyle \Gamma (1,x)=e^{-x}}" loading="lazy">,</li>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (1,x)=1-e^{-x}}</annotation>

</semantics>

</math><img src="./a85b6a9998af36e6cb6dab3761b9a4477cbf3851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.234ex; height:3.009ex;" alt="{\displaystyle \gamma (1,x)=1-e^{-x}}" loading="lazy">,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (0,x)=-\operatorname {Ei} (-x)}</annotation>

</semantics>

</math><img src="./411b5bcd20b40351eda0ebb6f05e11b662709cf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.259ex; height:2.843ex;" alt="{\displaystyle \Gamma (0,x)=-\operatorname {Ei} (-x)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy">,</li>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)=x^{s}\operatorname {E} _{1-s}(x)}</annotation>

</semantics>

</math><img src="./37fb109ef41fbe951036de902f6c403c6aa71a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.36ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,x)=x^{s}\operatorname {E} _{1-s}(x)}" loading="lazy">,</li>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma \left({\tfrac {1}{2}},x\right)={\sqrt {\pi }}\operatorname {erfc} \left({\sqrt {x}}\right)}</annotation>

</semantics>

</math><img src="./608631d652e49018fcb25ee27319b9fd004938eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.508ex; height:3.509ex;" alt="{\displaystyle \Gamma \left({\tfrac {1}{2}},x\right)={\sqrt {\pi }}\operatorname {erfc} \left({\sqrt {x}}\right)}" loading="lazy">,</li>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma \left({\tfrac {1}{2}},x\right)={\sqrt {\pi }}\operatorname {erf} \left({\sqrt {x}}\right)}</annotation>

</semantics>

</math><img src="./fe95ab8eb7e0c90fa34f21be4534b746ee1d1658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.285ex; height:3.509ex;" alt="{\displaystyle \gamma \left({\tfrac {1}{2}},x\right)={\sqrt {\pi }}\operatorname {erf} \left({\sqrt {x}}\right)}" loading="lazy">.</li></ul>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ei} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ei} }</annotation>

</semantics>

</math><img src="./6e9ef2bc3c11559409c3d53f635c610594e899a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.23ex; height:2.176ex;" alt="{\displaystyle \operatorname {Ei} }" loading="lazy"> is the <a href="Exponential_integral" title="Exponential integral">exponential integral</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{n}}</annotation>

</semantics>

</math><img src="./9a7fc04e2ace53adc62fa1ff7560e0f3efac8a5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle \operatorname {E} _{n}}" loading="lazy"> is the <a href="Exponential_integral#Relation_with_other_functions" title="Exponential integral">generalized exponential integral</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} }</annotation>

</semantics>

</math><img src="./1a8ee8591a5cc9057c9c646d1d07b848c1d68e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.153ex; width:2.809ex; height:2.176ex;" alt="{\displaystyle \operatorname {erf} }" loading="lazy"> is the <a href="Error_function" title="Error function">error function</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} }</annotation>

</semantics>

</math><img src="./7e0f7e177f6099d318fa89a322106c82af17a8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.176ex;" alt="{\displaystyle \operatorname {erfc} }" loading="lazy"> is the <a href="Complementary_error_function" class="mw-redirect" title="Complementary error function">complementary error function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)}</annotation>

</semantics>

</math><img src="./4cbd32d0b793f43ff11e30e0b575691959c68491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.723ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Asymptotic_behavior">Asymptotic behavior</h3></div>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\gamma (s,x)}{x^{s}}}\to {\frac {1}{s}}}</annotation>

</semantics>

</math><img src="./5f0be5f495e88957d397eeaee3b046d5f637593e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.974ex; height:5.676ex;" alt="{\displaystyle {\frac {\gamma (s,x)}{x^{s}}}\to {\frac {1}{s}}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to 0}</annotation>

</semantics>

</math><img src="./198f68f84d9d1c77e6d312f0c99bd0397e5ee49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\to 0}" loading="lazy">,</li>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma (s,x)}{x^{s}}}\to -{\frac {1}{s}}}</annotation>

</semantics>

</math><img src="./dd38c23a50132927ba3f08ca1924cb20c932c28b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.973ex; height:5.676ex;" alt="{\displaystyle {\frac {\Gamma (s,x)}{x^{s}}}\to -{\frac {1}{s}}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to 0}</annotation>

</semantics>

</math><img src="./198f68f84d9d1c77e6d312f0c99bd0397e5ee49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\to 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)<0}</annotation>

</semantics>

</math><img src="./15945d4d5b34abfaaa1468884e4df3e9ead81532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)<0}" loading="lazy"> (for real s, the error of Γ(s, x) ~ −xs / s is on the order of O(xmin{s + 1, 0}) if s ≠ −1 and O(ln(x)) if s = −1),</li>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)\sim \Gamma (s)-\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{s+n}}{n!(s+n)}}}</annotation>

</semantics>

</math><img src="./d562826ab6dfe0dc4be3652ddc968d7e8b829204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.373ex; height:6.843ex;" alt="{\displaystyle \Gamma (s,x)\sim \Gamma (s)-\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{s+n}}{n!(s+n)}}}" loading="lazy"> as an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to 0^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to 0^{+}}</annotation>

</semantics>

</math><img src="./7e10dcb09f3dc009185c645e27c0d670080d5db6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.617ex; height:2.509ex;" alt="{\displaystyle x\to 0^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\neq 0,-1,-2,\dots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\neq 0,-1,-2,\dots }</annotation>

</semantics>

</math><img src="./ce943ba288b4ae740b2b4754920e0e429d871bb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.118ex; height:2.676ex;" alt="{\displaystyle s\neq 0,-1,-2,\dots }" loading="lazy">.<a href="#cite_note-auto-13">[13]</a></li>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (-N,x)\sim C_{N}+{\frac {(-1)^{N+1}}{N!}}\ln x-\sum _{n=0,n\neq N}^{\infty }(-1)^{n}{\frac {x^{n-N}}{n!(n-N)}}}</annotation>

</semantics>

</math><img src="./a22a41d6f8f1669531e970f49eebbf243539cebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:59.333ex; height:7.509ex;" alt="{\displaystyle \Gamma (-N,x)\sim C_{N}+{\frac {(-1)^{N+1}}{N!}}\ln x-\sum _{n=0,n\neq N}^{\infty }(-1)^{n}{\frac {x^{n-N}}{n!(n-N)}}}" loading="lazy"> as an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to 0^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to 0^{+}}</annotation>

</semantics>

</math><img src="./7e10dcb09f3dc009185c645e27c0d670080d5db6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.617ex; height:2.509ex;" alt="{\displaystyle x\to 0^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1,2,\dots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N=1,2,\dots }</annotation>

</semantics>

</math><img src="./e9c7a0edd448735642461262a04bd71d8b752908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.278ex; height:2.509ex;" alt="{\displaystyle N=1,2,\dots }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{N}={\frac {(-1)^{N+1}}{N!}}\left(\gamma -\displaystyle \sum _{n=1}^{N}{\frac {1}{n}}\right)}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

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</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{N}={\frac {(-1)^{N+1}}{N!}}\left(\gamma -\displaystyle \sum _{n=1}^{N}{\frac {1}{n}}\right)}</annotation>

</semantics>

</math><img src="./21d2099b138f3ebfd731040963e59efeae18bbb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.866ex; height:7.509ex;" alt="{\textstyle C_{N}={\frac {(-1)^{N+1}}{N!}}\left(\gamma -\displaystyle \sum _{n=1}^{N}{\frac {1}{n}}\right)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is the <a href="Euler-Mascheroni_constant" class="mw-redirect" title="Euler-Mascheroni constant">Euler-Mascheroni constant</a>.<a href="#cite_note-auto-13">[13]</a></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)\to \Gamma (s)}</annotation>

</semantics>

</math><img src="./a6f2ceed1926930f230b3d9d41ed630670831be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.492ex; height:2.843ex;" alt="{\displaystyle \gamma (s,x)\to \Gamma (s)}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to \infty }</annotation>

</semantics>

</math><img src="./eda2caf97ec29f30d5f0c0cd7135393361efc020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\to \infty }" loading="lazy">,</li>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\Gamma (s,x)}{x^{s-1}e^{-x}}}\to 1}</annotation>

</semantics>

</math><img src="./19eafe5e9f89862bd076f17937209fd25f7033ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.581ex; height:6.009ex;" alt="{\displaystyle {\frac {\Gamma (s,x)}{x^{s-1}e^{-x}}}\to 1}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to \infty }</annotation>

</semantics>

</math><img src="./eda2caf97ec29f30d5f0c0cd7135393361efc020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\to \infty }" loading="lazy">,</li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)\sim z^{s-1}e^{-z}\sum _{k=0}{\frac {\Gamma (s)}{\Gamma (s-k)}}z^{-k}}</annotation>

</semantics>

</math><img src="./5ee5942cb632debf4302f790b0ef3b0cd12d0882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.958ex; height:7.009ex;" alt="{\displaystyle \Gamma (s,z)\sim z^{s-1}e^{-z}\sum _{k=0}{\frac {\Gamma (s)}{\Gamma (s-k)}}z^{-k}}" loading="lazy"> as an <a href="Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\to \infty }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |z|\to \infty }</annotation>

</semantics>

</math><img src="./878409f5f9775844a8d31df9dd60e41eadd1f889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.32ex; height:2.843ex;" alt="{\displaystyle |z|\to \infty }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\arg z\right|<{\tfrac {3}{2}}\pi }">

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|\arg z\right|<{\tfrac {3}{2}}\pi }</annotation>

</semantics>

</math><img src="./7cbe26de6c9c09d139a58c804dd76c382f56fdaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.094ex; height:3.509ex;" alt="{\displaystyle \left|\arg z\right|<{\tfrac {3}{2}}\pi }" loading="lazy">.<a href="#cite_note-14">[14]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Evaluation_formulae">Evaluation formulae</h2></div>

The lower gamma function can be evaluated using the power series expansion:<a href="#cite_note-15">[15]</a>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

</mrow>

<mrow>

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</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mrow>

</munderover>

<mfrac>

<msup>

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</msup>

<msup>

<mover>

<mrow>

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</mover>

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</msup>

</mfrac>

</mstyle>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)=\sum _{k=0}^{\infty }{\frac {z^{s}e^{-z}z^{k}}{s(s+1)\dots (s+k)}}=z^{s}e^{-z}\sum _{k=0}^{\infty }{\dfrac {z^{k}}{s^{\overline {k+1}}}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{\overline {k+1}}}">

<msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s^{\overline {k+1}}}</annotation>

</semantics>

</math><img src="./f8ba8609ecec0bc61cccdeb1e8c5b6274bffc34d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.373ex; height:3.176ex;" alt="{\displaystyle s^{\overline {k+1}}}" loading="lazy"> is the <a href="Falling_and_rising_factorials" title="Falling and rising factorials">Pochhammer symbol</a>.

An alternative expansion is

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

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</msup>

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<mrow>

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<mfrac>

<msup>

</mrow>

</msup>

<mrow>

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</mfrac>

</mrow>

<mfrac>

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}{\frac {z^{s+k}}{s+k}}={\frac {z^{s}}{s}}M(s,s+1,-z),}</annotation>

</semantics>

</math>

where M is Kummer's <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a>.

<div class="mw-heading mw-heading3"><h3 id="Connection_with_Kummer's_confluent_hypergeometric_function">Connection with Kummer's confluent hypergeometric function</h3></div>

When the real part of z is positive,

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)=s^{-1}z^{s}e^{-z}M(1,s+1,z)}</annotation>

</semantics>

</math>

where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(1,s+1,z)=1+{\frac {z}{(s+1)}}+{\frac {z^{2}}{(s+1)(s+2)}}+{\frac {z^{3}}{(s+1)(s+2)(s+3)}}+\cdots }">

<mfrac>

<mrow>

</mrow>

</mfrac>

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<mfrac>

<msup>

</mrow>

</msup>

<mrow>

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<mfrac>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(1,s+1,z)=1+{\frac {z}{(s+1)}}+{\frac {z^{2}}{(s+1)(s+2)}}+{\frac {z^{3}}{(s+1)(s+2)(s+3)}}+\cdots }</annotation>

</semantics>

</math> has an infinite radius of convergence.

Again with <a href="Confluent_hypergeometric_functions" class="mw-redirect" title="Confluent hypergeometric functions">confluent hypergeometric functions</a> and employing Kummer's identity,

<mtr>

<mtd>

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</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Gamma (s,z)&=e^{-z}U(1-s,1-s,z)={\frac {z^{s}e^{-z}}{\Gamma (1-s)}}\int _{0}^{\infty }{\frac {e^{-u}}{u^{s}(z+u)}}du\\&=e^{-z}z^{s}U(1,1+s,z)=e^{-z}\int _{0}^{\infty }e^{-u}(z+u)^{s-1}du=e^{-z}z^{s}\int _{0}^{\infty }e^{-zu}(1+u)^{s-1}du.\end{aligned}}}</annotation>

</semantics>

</math>

For the actual computation of numerical values, <a href="Gauss's_continued_fraction" title="Gauss's continued fraction">Gauss's continued fraction</a> provides a useful expansion:

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,z)={\cfrac {z^{s}e^{-z}}{s-{\cfrac {sz}{s+1+{\cfrac {z}{s+2-{\cfrac {(s+1)z}{s+3+{\cfrac {2z}{s+4-{\cfrac {(s+2)z}{s+5+{\cfrac {3z}{s+6-\ddots }}}}}}}}}}}}}}.}</annotation>

</semantics>

</math>

This continued fraction converges for all complex z, provided only that s is not a negative integer.

The upper gamma function has the continued fraction<a href="#cite_note-16">[16]</a>

<mfrac>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)={\cfrac {z^{s}e^{-z}}{z+{\cfrac {1-s}{1+{\cfrac {1}{z+{\cfrac {2-s}{1+{\cfrac {2}{z+{\cfrac {3-s}{1+\ddots }}}}}}}}}}}}}</annotation>

</semantics>

</math>

and

<mfrac>

<mrow>

</mpadded>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)={\cfrac {z^{s}e^{-z}}{1+z-s+{\cfrac {s-1}{3+z-s+{\cfrac {2(s-2)}{5+z-s+{\cfrac {3(s-3)}{7+z-s+{\cfrac {4(s-4)}{9+z-s+\ddots }}}}}}}}}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Multiplication_theorem">Multiplication theorem</h3></div>

The following <a href="Multiplication_theorem" title="Multiplication theorem">multiplication theorem</a> holds true:

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,z)={\frac {1}{t^{s}}}\sum _{i=0}^{\infty }{\frac {\left(1-{\frac {1}{t}}\right)^{i}}{i!}}\Gamma (s+i,tz)=\Gamma (s,tz)-(tz)^{s}e^{-tz}\sum _{i=1}^{\infty }{\frac {\left({\frac {1}{t}}-1\right)^{i}}{i}}L_{i-1}^{(s-i)}(tz).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Software_implementation">Software implementation</h3></div>

The incomplete gamma functions are available in various of the <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>.

Even if unavailable directly, however, incomplete function values can be calculated using functions commonly included in <a href="Spreadsheet" title="Spreadsheet">spreadsheets</a> (and computer algebra packages). In <a href="Microsoft_Excel" title="Microsoft Excel">Excel</a>, for example, these can be calculated using the <a href="Gamma_function" title="Gamma function">gamma function</a> combined with the <a href="Gamma_distribution" title="Gamma distribution">gamma distribution</a> function.

<ul><li>The lower incomplete function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (s,x)}</annotation>

</semantics>

</math><img src="./7c21837ee68b66ff0e6bfd1df23027b4510e2578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.526ex; height:2.843ex;" alt="{\displaystyle \gamma (s,x)}" loading="lazy"> <code> = EXP(GAMMALN(s))*GAMMA.DIST(x,s,1,TRUE)</code>.</li>

<li>The upper incomplete function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)}</annotation>

</semantics>

</math><img src="./de56b620724ff26935177bd16ff2d296baea4f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,x)}" loading="lazy"> <code> = EXP(GAMMALN(s))*(1-GAMMA.DIST(x,s,1,TRUE))</code>.</li></ul>

These follow from the definition of the <a href="Gamma_distribution#Cumulative_distribution_function" title="Gamma distribution">gamma distribution's cumulative distribution function</a>.

In <a href="Python_(programming_language)" title="Python (programming language)">Python</a>, the Scipy library provides implementations of incomplete gamma functions under <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">scipy.special</code>, however, it does not support negative values for the first argument. The function <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">gammainc</code> from the mpmath library supports all complex arguments.

<div class="mw-heading mw-heading2"><h2 id="Regularized_gamma_functions_and_Poisson_random_variables">Regularized gamma functions and Poisson random variables</h2></div>

Two related functions are the regularized gamma functions:

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P(s,x)&={\frac {\gamma (s,x)}{\Gamma (s)}},\\[1ex]Q(s,x)&={\frac {\Gamma (s,x)}{\Gamma (s)}}=1-P(s,x).\end{aligned}}}</annotation>

</semantics>

</math>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(s,x)}</annotation>

</semantics>

</math><img src="./fc561423e4143752c9d9a5533885a49c06c3af50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.009ex; height:2.843ex;" alt="{\displaystyle P(s,x)}" loading="lazy"> is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> for <a href="Gamma_distribution" title="Gamma distribution">gamma random variables</a> with <a href="Shape_parameter" title="Shape parameter">shape parameter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> and <a href="Scale_parameter" title="Scale parameter">scale parameter</a> 1.

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is an integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(s+1,\lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(s+1,\lambda )}</annotation>

</semantics>

</math><img src="./c20a5707efb889c0425ab0d6b4e02e922a6320e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.13ex; height:2.843ex;" alt="{\displaystyle Q(s+1,\lambda )}" loading="lazy"> is the cumulative distribution function for <a href="Poisson_random_variable" class="mw-redirect" title="Poisson random variable">Poisson random variables</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Poi} (\lambda )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Poi} (\lambda )}</annotation>

</semantics>

</math><img src="./211ecc86b32f649090c2f1e4847b78dcb8337179.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.557ex; height:2.843ex;" alt="{\displaystyle \mathrm {Poi} (\lambda )}" loading="lazy"> random variable then

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pr(X\leq s)=\sum _{i\leq s}e^{-\lambda }{\frac {\lambda ^{i}}{i!}}={\frac {\Gamma (s+1,\lambda )}{\Gamma (s+1)}}=Q(s+1,\lambda ).}</annotation>

</semantics>

</math>

This formula can be derived by repeated integration by parts.

In the context of the <a href="Stable_count_distribution" title="Stable count distribution">stable count distribution</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> parameter can be regarded as inverse of Lévy's stability parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">:

</mrow>

</mrow>

</msubsup>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</mrow>

</mrow>

</msub>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(s,x)=\int _{0}^{\infty }e^{\left(-{x^{s}}/{\nu }\right)}\,{\mathfrak {N}}_{{1}/{s}}\left(\nu \right)\,d\nu ,\quad (s>1)}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}</annotation>

</semantics>

</math><img src="./16503339cce78afe1b7b86dcf6d064fb7f34b979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {N}}_{\alpha }(\nu )}" loading="lazy"> is a standard stable count distribution of shape <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1/s<1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =1/s<1}</annotation>

</semantics>

</math><img src="./5729c544ebfa58cd94a4f2584d0b673f58e9d856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.262ex; height:2.843ex;" alt="{\displaystyle \alpha =1/s<1}" loading="lazy">.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(s,x)}</annotation>

</semantics>

</math><img src="./fc561423e4143752c9d9a5533885a49c06c3af50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.009ex; height:2.843ex;" alt="{\displaystyle P(s,x)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(s,x)}</annotation>

</semantics>

</math><img src="./2c1422397a25f18de9a4877ed6291f9144f222b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.102ex; height:2.843ex;" alt="{\displaystyle Q(s,x)}" loading="lazy"> are implemented as <code>gammainc</code><a href="#cite_note-17">[17]</a> and <code>gammaincc</code><a href="#cite_note-18">[18]</a> in <a href="Scipy" class="mw-redirect" title="Scipy">scipy</a>.

<div class="mw-heading mw-heading2"><h2 id="Derivatives">Derivatives</h2></div>

Using the integral representation above, the derivative of the upper incomplete gamma function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (s,x)}</annotation>

</semantics>

</math><img src="./de56b620724ff26935177bd16ff2d296baea4f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:2.843ex;" alt="{\displaystyle \Gamma (s,x)}" loading="lazy"> with respect to x is

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Gamma (s,x)}{\partial x}}=-x^{s-1}e^{-x}}</annotation>

</semantics>

</math>

The derivative with respect to its first argument <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is given by<a href="#cite_note-19">[19]</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Gamma (s,x)}{\partial s}}=\ln x\Gamma (s,x)+x\,T(3,s,x)}</annotation>

</semantics>

</math>

and the second derivative by

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\Gamma (s,x)}{\partial s^{2}}}=\ln ^{2}x\Gamma (s,x)+2x\left[\ln x\,T(3,s,x)+T(4,s,x)\right]}</annotation>

</semantics>

</math>

where the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(m,s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(m,s,x)}</annotation>

</semantics>

</math><img src="./cd80c1b3744b6d32707d9e8e53897741853aafc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:2.843ex;" alt="{\displaystyle T(m,s,x)}" loading="lazy"> is a special case of the <a href="Meijer_G-function" title="Meijer G-function">Meijer G-function</a>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(m,s,x)=G_{m-1,\,m}^{\,m,\,0}\!\left(\left.{\begin{matrix}0,0,\dots ,0\\s-1,-1,\dots ,-1\end{matrix}}\;\right|\,x\right).}</annotation>

</semantics>

</math>

This particular special case has internal closure properties of its own because it can be used to express all successive derivatives. In general,

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{m}\Gamma (s,x)}{\partial s^{m}}}=\ln ^{m}x\Gamma (s,x)+mx\,\sum _{n=0}^{m-1}P_{n}^{m-1}\ln ^{m-n-1}x\,T(3+n,s,x)}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{j}^{n}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{j}^{n}}</annotation>

</semantics>

</math><img src="./24ff747ab5e82f587d7dc43e2f9c9f69ceff6ee4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.04ex; height:3.176ex;" alt="{\displaystyle P_{j}^{n}}" loading="lazy"> is the <a href="Permutation" title="Permutation">permutation</a> defined by the <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>:

</mrow>

</mrow>

</msubsup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{j}^{n}={\binom {n}{j}}j!={\frac {n!}{(n-j)!}}.}</annotation>

</semantics>

</math>

All such derivatives can be generated in succession from:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial T(m,s,x)}{\partial s}}=\ln x~T(m,s,x)+(m-1)T(m+1,s,x)}</annotation>

</semantics>

</math>

and

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial T(m,s,x)}{\partial x}}=-{\frac {T(m-1,s,x)+T(m,s,x)}{x}}}</annotation>

</semantics>

</math>

This function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(m,s,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(m,s,x)}</annotation>

</semantics>

</math><img src="./cd80c1b3744b6d32707d9e8e53897741853aafc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:2.843ex;" alt="{\displaystyle T(m,s,x)}" loading="lazy"> can be computed from its series representation valid for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|<1}">

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<annotation encoding="application/x-tex">{\displaystyle |z|<1}</annotation>

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</math><img src="./e1c0fa57b899b653a3823f85f43fd666309c09b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|<1}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle T(m,s,z)=-{\frac {\left(-1\right)^{m-1}}{(m-2)!}}\left.{\frac {d^{m-2}}{dt^{m-2}}}\left[\Gamma (s-t)z^{t-1}\right]\right|_{t=0}+\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}z^{s-1+n}}{n!\left(-s-n\right)^{m-1}}}}</annotation>

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with the understanding that s is not a negative integer or zero. In such a case, one must use a limit. Results for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle |z|\geq 1}</annotation>

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</math><img src="./22478941c4982a11952c4b21e2944202f4936eee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.643ex; height:2.843ex;" alt="{\displaystyle |z|\geq 1}" loading="lazy"> can be obtained by <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a>. Some special cases of this function can be simplified. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(2,s,x)=\Gamma (s,x)/x}">

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<annotation encoding="application/x-tex">{\displaystyle T(2,s,x)=\Gamma (s,x)/x}</annotation>

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</math><img src="./6d7be991a14d56c2cb0f54bb114658ac9e208635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.403ex; height:2.843ex;" alt="{\displaystyle T(2,s,x)=\Gamma (s,x)/x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,T(3,1,x)=\mathrm {E} _{1}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle x\,T(3,1,x)=\mathrm {E} _{1}(x)}</annotation>

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</math><img src="./30d3a5c5a7bc87daf997f4ac728753d4bc4257bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.759ex; height:2.843ex;" alt="{\displaystyle x\,T(3,1,x)=\mathrm {E} _{1}(x)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} _{1}(x)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} _{1}(x)}</annotation>

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</math><img src="./00570a6e879e5411b30de2e59fce176f3903e6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.776ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} _{1}(x)}" loading="lazy"> is the <a href="Exponential_integral" title="Exponential integral">Exponential integral</a>. These derivatives and the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(m,s,x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T(m,s,x)}</annotation>

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</math><img src="./cd80c1b3744b6d32707d9e8e53897741853aafc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.974ex; height:2.843ex;" alt="{\displaystyle T(m,s,x)}" loading="lazy"> provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a>

For example,

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<annotation encoding="application/x-tex">{\displaystyle \int _{x}^{\infty }{\frac {t^{s-1}\ln ^{m}t}{e^{t}}}dt={\frac {\partial ^{m}}{\partial s^{m}}}\int _{x}^{\infty }{\frac {t^{s-1}}{e^{t}}}dt={\frac {\partial ^{m}}{\partial s^{m}}}\Gamma (s,x)}</annotation>

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</math>

This formula can be further inflated or generalized to a huge class of <a href="Laplace_transform" title="Laplace transform">Laplace transforms</a> and <a href="Mellin_transform" title="Mellin transform">Mellin transforms</a>. When combined with a <a href="Computer_algebra_system" title="Computer algebra system">computer algebra system</a>, the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see <a href="Symbolic_integration" title="Symbolic integration">Symbolic integration</a> for more details).

<div class="mw-heading mw-heading2"><h2 id="Indefinite_and_definite_integrals">Indefinite and definite integrals</h2></div>

The following indefinite integrals are readily obtained using <a href="Integration_by_parts" title="Integration by parts">integration by parts</a> (with the <a href="Constant_of_integration" title="Constant of integration">constant of integration</a> omitted in both cases):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int x^{b-1}\gamma (s,x)\,dx&={\frac {1}{b}}\left(x^{b}\gamma (s,x)-\gamma (s+b,x)\right),\\[1ex]\int x^{b-1}\Gamma (s,x)\,dx&={\frac {1}{b}}\left(x^{b}\Gamma (s,x)-\Gamma (s+b,x)\right).\end{aligned}}}</annotation>

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</math>

The lower and the upper incomplete gamma function are connected via the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>:

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }{\frac {\gamma \left({\frac {s}{2}},z^{2}\pi \right)}{(z^{2}\pi )^{\frac {s}{2}}}}e^{-2\pi ikz}dz={\frac {\Gamma \left({\frac {1-s}{2}},k^{2}\pi \right)}{(k^{2}\pi )^{\frac {1-s}{2}}}}.}</annotation>

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This follows, for example, by suitable specialization of (<a href="#CITEREFGradshteynRyzhikGeronimusTseytlin2015">Gradshteyn et al. 2015</a>, §7.642).

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/8.2">"DLMF: §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions"</a>. dlmf.nist.gov.</cite>

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<li id="cite_note-auto2-2">^ <a href="#cite_ref-auto2_2-0">a</a> <a href="#cite_ref-auto2_2-1">b</a> <a href="#cite_ref-auto2_2-2">c</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/8.8">"DLMF: §8.8 Recurrence Relations and Derivatives ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions"</a>. dlmf.nist.gov.</cite>

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<li id="cite_note-class_notes-3">^ <a href="#cite_ref-class_notes_3-0">a</a> <a href="#cite_ref-class_notes_3-1">b</a> <cite id="CITEREFDonald_E._Marshall2009" class="citation web cs1">Donald E. Marshall (Autumn 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110516005152/http://www.math.washington.edu/~marshall/math_534/Notes.pdf">"Complex Analysis"</a> (PDF). Math 534 (student handout). University of Washington. Theorem 3.9 on p.56. Archived from <a rel="nofollow" class="external text" href="http://www.math.washington.edu/~marshall/math_534/Notes.pdf">the original</a> (PDF) on 16 May 2011. Retrieved 23 April 2011.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFPaul_Garrett" class="citation web cs1">Paul Garrett. <a rel="nofollow" class="external text" href="https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf">"Hartogs' Theorem: separate analyticity implies joint"</a> (PDF). cse.umn.edu. Retrieved 21 December 2023.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFC._Teleman" class="citation web cs1">C. Teleman. <a rel="nofollow" class="external text" href="http://math.berkeley.edu/~teleman/math/Riemann.pdf">"Riemann Surfaces"</a> (PDF). berkeley.edu. Retrieved 21 December 2023.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/5.2">"DLMF: §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function"</a>. dlmf.nist.gov.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/4.4">"DLMF: §4.4 Special Values and Limits ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions"</a>. dlmf.nist.gov.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/8.4">"DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions"</a>. dlmf.nist.gov.</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/8.4.E15">"DLMF: 8.4 Special Values"</a>.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">"Incomplete Gamma Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite> (equation 2)

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<li id="cite_note-auto-13">^ <a href="#cite_ref-auto_13-0">a</a> <a href="#cite_ref-auto_13-1">b</a> <cite id="CITEREFBender_&_Orszag1978" class="citation book cs1">Bender & Orszag (1978). Advanced Mathematical Methods for Scientists and Engineers. Springer. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978amms.book.....B">1978amms.book.....B</a>.</cite>

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<li id="cite_note-16"><a href="#cite_ref-16">^</a> Abramowitz and Stegun <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_263.htm">p. 263, 6.5.31</a>

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammainc.html#scipy.special.gammainc">"scipy.special.gammainc — SciPy v1.11.4 Manual"</a>. docs.scipy.org.</cite>

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<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="Keith_Geddes" title="Keith Geddes">K.O. Geddes</a>, M.L. Glasser, R.A. Moore and T.C. Scott, Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions, AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp. 149–165, <a rel="nofollow" class="external autonumber" href="https://doi.org/10.1007%2FBF01810298">[1]</a>

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<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFMathar2009" class="citation arxiv cs1">Mathar (2009). "Numerical Evaluation of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0912.3844">0912.3844</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CA">math.CA</a>].</cite>, App B

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. "Chapter 6.5". <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_260.htm">"Incomplete Gamma function"</a>.</cite> §6.5.</li>

<li><cite id="CITEREFAllasiaBesenghi1986" class="citation journal cs1">Allasia, Giampietro; Besenghi, Renata (1986). "Numerical calculation of incomplete gamma functions by the trapezoidal rule". Numer. Math. 50 (4): 419–428. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01396662">10.1007/BF01396662</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121964300">121964300</a>.</cite></li>

<li><cite id="CITEREFAmore2005" class="citation journal cs1">Amore, Paolo (2005). "Asymptotic and exact series representations for the incomplete Gamma function". Europhys. Lett. 71 (1): 1–7. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math-ph/0501019">math-ph/0501019</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005EL.....71....1A">2005EL.....71....1A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1209%2Fepl%2Fi2005-10066-6">10.1209/epl/i2005-10066-6</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2170316">2170316</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1921569">1921569</a>.</cite></li>

<li>G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)</li>

<li><cite id="CITEREFDiDonatoMorris,_Jr.1986" class="citation journal cs1">DiDonato, Armido R.; Morris, Jr., Alfred H. (December 1986). "Computation of the incomplete gamma function ratios and their inverse". ACM Transactions on Mathematical Software. 12 (4): 377–393. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F22721.23109">10.1145/22721.23109</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14351930">14351930</a>.</cite></li>

<li><cite id="CITEREFBarakat1961" class="citation journal cs1">Barakat, Richard (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-1961-0128058-1">"Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials"</a>. Math. Comp. 15 (73): 7–11. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0025-5718-1961-0128058-1">10.1090/s0025-5718-1961-0128058-1</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0128058">0128058</a>.</cite></li>

<li><cite id="CITEREFCarskyPolasek1998" class="citation journal cs1">Carsky, Petr; Polasek, Martin (1998). "Incomplete Gamma F_m(x) functions for real and complex arguments". J. Comput. Phys. 143 (1): 259–265. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998JCoPh.143..259C">1998JCoPh.143..259C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjcph.1998.5975">10.1006/jcph.1998.5975</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1624704">1624704</a>.</cite></li>

<li><cite id="CITEREFChaudhryZubair1995" class="citation journal cs1">Chaudhry, M. Aslam; Zubair, S. M. (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2894%2900026-w">"On the decomposition of generalized incomplete Gamma functions with applications to Fourier transforms"</a>. J. Comput. Appl. Math. 59 (101): 253–284. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2894%2900026-w">10.1016/0377-0427(94)00026-w</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1346414">1346414</a>.</cite></li>

<li><cite id="CITEREFDiDonatoMorris,_Jr.1987" class="citation journal cs1">DiDonato, Armido R.; Morris, Jr., Alfred H. (September 1987). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1236180">"ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse"</a>. ACM Transactions on Mathematical Software. 13 (3): 318–319. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F29380.214348">10.1145/29380.214348</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:19902932">19902932</a>.</cite> (See also <a rel="nofollow" class="external text" href="http://www.netlib.org/toms/654">www.netlib.org/toms/654</a>).</li>

<li><cite id="CITEREFFrüchtlOtto1994" class="citation journal cs1">Früchtl, H.; Otto, P. (1994). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F198429.198432">"A new algorithm for the evaluation of the incomplete Gamma Function on vector computers"</a>. ACM Trans. Math. Softw. 20 (4): 436–446. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F198429.198432">10.1145/198429.198432</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16737306">16737306</a>.</cite></li>

<li><cite id="CITEREFGautschi1998" class="citation journal cs1">Gautschi, Walter (1998). "The incomplete gamma function since Tricomi". Atti Convegni Lincei. 147: 203–237. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1737497">1737497</a>.</cite></li>

<li><cite id="CITEREFGautschi1999" class="citation journal cs1">Gautschi, Walter (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F305658.305717">"A Note on the recursive calculation of Incomplete Gamma Functions"</a>. ACM Trans. Math. Softw. 25 (1): 101–107. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F305658.305717">10.1145/305658.305717</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697463">1697463</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:36469885">36469885</a>.</cite></li>

<li><cite id="CITEREFGradshteynRyzhikGeronimusTseytlin2015" class="citation book cs1"><a href="Izrail_Solomonovich_Gradshteyn" class="mw-redirect" title="Izrail Solomonovich Gradshteyn">Gradshteyn, Izrail Solomonovich</a>; <a href="Iosif_Moiseevich_Ryzhik" class="mw-redirect" title="Iosif Moiseevich Ryzhik">Ryzhik, Iosif Moiseevich</a>; <a href="Yuri_Veniaminovich_Geronimus" class="mw-redirect" title="Yuri Veniaminovich Geronimus">Geronimus, Yuri Veniaminovich</a>; <a href="Michail_Yulyevich_Tseytlin" class="mw-redirect" title="Michail Yulyevich Tseytlin">Tseytlin, Michail Yulyevich</a>; Jeffrey, Alan (2015) [October 2014]. "8.35.". In Zwillinger, Daniel; <a href="Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (eds.). <a href="Gradshteyn_and_Ryzhik" title="Gradshteyn and Ryzhik">Table of Integrals, Series, and Products</a>. Translated by Scripta Technica, Inc. (8 ed.). <a href="Academic_Press%2C_Inc." class="mw-redirect" title="Academic Press, Inc.">Academic Press, Inc.</a> pp. 908–911. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-384933-5</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2014010276">2014010276</a>.</cite></li>

<li><cite id="CITEREFJonesThron1985" class="citation journal cs1 cs1-prop-long-vol">Jones, William B.; Thron, W. J. (1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2885%2990034-2">"On the computation of incomplete gamma functions in the complex domain"</a>. J. Comput. Appl. Math. 12–13: 401–417. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2885%2990034-2">10.1016/0377-0427(85)90034-2</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0793971">0793971</a>.</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Incomplete_gamma-function">"Incomplete gamma-function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

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<li><cite id="CITEREFParis2010" class="citation cs2">Paris, R. B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/8">"Incomplete gamma function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li>

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<li><cite id="CITEREFvan_DeunCools2006" class="citation journal cs1">van Deun, Joris; Cools, Ronald (2006). "A stable recurrence for the incomplete gamma function with imaginary second argument". Numer. Math. 104 (4): 445–456. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00211-006-0026-1">10.1007/s00211-006-0026-1</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2249673">2249673</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43780150">43780150</a>.</cite></li>

<li><cite id="CITEREFWinitzki2003" class="citation conference cs1">Winitzki, Serge (2003). "Computing the Incomplete Gamma Function to Arbitrary Precision". In Vipin Kumar; <a href="Marina_Gavrilova" title="Marina Gavrilova">Marina L. Gavrilova</a>; Chih Jeng Kenneth Tan; Pierre L'Ecuyer (eds.). Computational Science and Its Applications — ICSSA 2003. International Conference on Computational Science and Its Applications, Montreal, Canada, May 18–21, 2003, Proceedings, Part I. Lecture Notes in Computer Science. Vol. 2667. pp. 790–798. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44839-x_83">10.1007/3-540-44839-x_83</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-40155-1</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2110953">2110953</a>.</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">"Incomplete Gamma Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle Q(a,x)}</annotation>

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</math><img src="./9172790934463388ffadbd6210824133db6a8fd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.241ex; height:2.843ex;" alt="{\displaystyle Q(a,x)}" loading="lazy"> — <a rel="nofollow" class="external text" href="https://www.danielsoper.com/statcalc/calculator.aspx?id=34">Regularized Upper Incomplete Gamma Function Calculator</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \gamma (a,x)}</annotation>

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</math><img src="./bde5439ab55d3a653baf8b580a3a2cd6d35dbea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.665ex; height:2.843ex;" alt="{\displaystyle \gamma (a,x)}" loading="lazy"> — <a rel="nofollow" class="external text" href="https://www.danielsoper.com/statcalc/calculator.aspx?id=24">Lower Incomplete Gamma Function Calculator</a></li>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (a,x)}</annotation>

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</math><img src="./90d612673ea5d2db902008df9828ef31c03da065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.856ex; height:2.843ex;" alt="{\displaystyle \Gamma (a,x)}" loading="lazy"> — <a rel="nofollow" class="external text" href="https://www.danielsoper.com/statcalc/calculator.aspx?id=23">Upper Incomplete Gamma Function Calculator</a></li>

<li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/GammaBetaErf/Gamma3/">formulas and identities of the Incomplete Gamma Function</a> functions.wolfram.com</li></ul></div><div><div class="zim-footer">

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