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Entire function

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In <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, an entire function, also called an integral function, is a complex-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that is <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> on the whole <a href="Complex_plane" title="Complex plane">complex plane</a>. Typical examples of entire functions are <a href="Polynomial" title="Polynomial">polynomials</a> and the <a href="Exponential_function" title="Exponential function">exponential function</a>, and any finite sums, products and compositions of these, such as the <a href="Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> <a href="Sine" class="mw-redirect" title="Sine">sine</a> and <a href="Cosine" class="mw-redirect" title="Cosine">cosine</a> and their <a href="Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic counterparts</a> <a href="Hyperbolic_sine" class="mw-redirect" title="Hyperbolic sine">sinh</a> and <a href="Hyperbolic_cosine" class="mw-redirect" title="Hyperbolic cosine">cosh</a>, as well as <a href="Derivative" title="Derivative">derivatives</a> and <a href="Integral" title="Integral">integrals</a> of entire functions such as the <a href="Error_function" title="Error function">error function</a>. If an entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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

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</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> has a

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

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</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)/(z-w)}">

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

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</math><img src="./2cc309277810b7b9241e1fb09f23ba95d5300ad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.74ex; height:2.843ex;" alt="{\displaystyle f(z)/(z-w)}" loading="lazy">, taking the limit value at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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

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</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy">, is an entire function. On the other hand, the <a href="Natural_logarithm" title="Natural logarithm">natural logarithm</a>, the <a href="Reciprocal_function" class="mw-redirect" title="Reciprocal function">reciprocal function</a>, and the <a href="Square_root" title="Square root">square root</a> are all not entire functions, nor can they be <a href="Analytic_continuation" title="Analytic continuation">continued analytically</a> to an entire function.

A <a href="Transcendental_function" title="Transcendental function">transcendental</a> entire function is an entire function that is not a polynomial.

Just as <a href="Meromorphic_function" title="Meromorphic function">meromorphic functions</a> can be viewed as a generalization of <a href="Rational_function" title="Rational function">rational fractions</a>, entire functions can be viewed as a generalization of polynomials. In particular, if for meromorphic functions one can generalize the factorization into simple fractions (the <a href="Mittag-Leffler's_theorem" title="Mittag-Leffler's theorem">Mittag-Leffler theorem</a> on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the <a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass theorem</a> on entire functions.

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

Every entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

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

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</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> can be represented as a single <a href="Power_series" title="Power series">power series</a>:

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<annotation encoding="application/x-tex">{\displaystyle \ f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\ }</annotation>

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that <a href="Convergence_(mathematics)" class="mw-redirect" title="Convergence (mathematics)">converges</a> everywhere in the complex plane, hence <a href="Compact_convergence" title="Compact convergence">uniformly on compact sets</a>. The <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> is infinite, which implies that

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<annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}=0\ }</annotation>

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or, equivalently,<a href="#cite_note-1">[a]</a>

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<annotation encoding="application/x-tex">{\displaystyle \ \lim _{n\to \infty }{\frac {\ln |a_{n}|}{n}}=-\infty ~.}</annotation>

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Any power series satisfying this criterion will represent an entire function.

If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> will be the complex conjugate of the value at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z~.}">

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

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</math><img src="./eed2c71d8bd94cf0be508084317dba86d58e6de6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.316ex; height:1.676ex;" alt="{\displaystyle z~.}" loading="lazy"> Such functions are sometimes called self-conjugate (the conjugate function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}(z),}">

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

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</math><img src="./9e26d70d53eb39f2460b354d33d3a1b19caeecc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.413ex; height:2.843ex;" alt="{\displaystyle F^{*}(z),}" loading="lazy"> being given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {F}}({\bar {z}})}">

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

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</math><img src="./81050547ce9a972cb6f8957585f87cef9f3e60e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.911ex; height:3.009ex;" alt="{\displaystyle {\bar {F}}({\bar {z}})}" loading="lazy">).<a href="#cite_note-FOOTNOTEBoas19541-2">[1]</a>

If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, <a href="Up_to" title="Up to">up to</a> an imaginary constant. For instance, if the real part is known in a <a href="Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighborhood</a> of zero, then we can find the coefficients for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

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

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</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"> from the following derivatives with respect to a real variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {\mathcal {Re}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f(r)\ \right\}&&\quad \mathrm {at} \quad r=0\\\operatorname {\mathcal {Im}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f\left(r\ e^{-{\frac {i\pi }{2n}}}\right)\ \right\}&&\quad \mathrm {at} \quad r=0\end{aligned}}}</annotation>

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(Likewise, if the imaginary part is known in a neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant.<a href="#cite_note-3">[b]</a>}

Note however that an entire function is not determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some <a href="Imaginary_number" title="Imaginary number">imaginary number</a>.

The <a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass factorization theorem</a> asserts that any entire function can be represented by a product involving its <a href="Zero_of_a_function" title="Zero of a function">zeroes</a> (or "roots").

The entire functions on the complex plane form an <a href="Integral_domain" title="Integral domain">integral domain</a> (in fact a <a href="Pr%C3%BCfer_domain" title="Prüfer domain">Prüfer domain</a>). They also form a <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="Unital_algebra" class="mw-redirect" title="Unital algebra">unital</a> <a href="Associative_algebra" title="Associative algebra">associative algebra</a> over the <a href="Complex_number" title="Complex number">complex numbers</a>.

<a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a> states that any <a href="Bounded_function" title="Bounded function">bounded</a> entire function must be constant.<a href="#cite_note-4">[c]</a>

As a consequence of Liouville's theorem, any function that is entire on the whole <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a><a href="#cite_note-5">[d]</a>

is constant. Thus any non-constant entire function must have a <a href="Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularity</a> at the complex <a href="Point_at_infinity" title="Point at infinity">point at infinity</a>, either a <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> for a polynomial or an <a href="Essential_singularity" title="Essential singularity">essential singularity</a> for a <a href="Transcendental_function" title="Transcendental function">transcendental</a> entire function. Specifically, by the <a href="Casorati%E2%80%93Weierstrass_theorem" title="Casorati–Weierstrass theorem">Casorati–Weierstrass theorem</a>, for any transcendental entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and any complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

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

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</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> there is a <a href="Sequence" title="Sequence">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z_{m})_{m\in \mathbb {N} }}">

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<annotation encoding="application/x-tex">{\displaystyle (z_{m})_{m\in \mathbb {N} }}</annotation>

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</math><img src="./929bc7644348dc5ff0c1f26be94f6ea6528b8eb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle (z_{m})_{m\in \mathbb {N} }}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle \ \lim _{m\to \infty }|z_{m}|=\infty ,\qquad {\text{and}}\qquad \lim _{m\to \infty }f(z_{m})=w~.}</annotation>

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</math><img src="./9f07dbaa7bd4eb445172bc92c5fa2c9bebc4aead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.965ex; height:3.843ex;" alt="{\displaystyle \ \lim _{m\to \infty }|z_{m}|=\infty ,\qquad {\text{and}}\qquad \lim _{m\to \infty }f(z_{m})=w~.}" loading="lazy"></dd></dl>

<a href="Picard_theorem" title="Picard theorem">Picard's little theorem</a> is a much stronger result: Any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a <a href="Lacunary_value" title="Lacunary value">lacunary value</a> of the function. The possibility of a lacunary value is illustrated by the <a href="Exponential_function" title="Exponential function">exponential function</a>, which never takes on the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">. One can take a suitable branch of the logarithm of an entire function that never hits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, so that this will also be an entire function (according to the <a href="Weierstrass_factorization_theorem" title="Weierstrass factorization theorem">Weierstrass factorization theorem</a>). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.

Liouville's theorem is a special case of the following statement:

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Theorem—Assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,}">

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

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</math><img src="./b466e90209f39c0c2caad1b11445824b82c2f536.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.089ex; height:2.509ex;" alt="{\displaystyle M,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is a non-negative integer. An entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> satisfying the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(z)|\leq M|z|^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle |f(z)|\leq M|z|^{n}}</annotation>

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</math><img src="./52458c75a7ce96ab071e24ac9608f9244e3965a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.611ex; height:3.009ex;" alt="{\displaystyle |f(z)|\leq M|z|^{n}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\geq R,}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7ce311afc5cf9dfee94fdb8c1231502fb1001e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.891ex; height:2.843ex;" alt="{\displaystyle |z|\geq R,}" loading="lazy"> is necessarily a polynomial, of <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n~.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n~.}</annotation>

</semantics>

</math><img src="./dbd0742bb5ffce87e1ee3b152830e1729ed0d0b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:1.676ex;" alt="{\displaystyle n~.}" loading="lazy"><a href="#cite_note-6">[e]</a>

Similarly, an entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> satisfying the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M|z|^{n}\leq |f(z)|}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M|z|^{n}\leq |f(z)|}</annotation>

</semantics>

</math><img src="./2d3560fbfa3185f6c2ce16be0daaf06d8de1db28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.611ex; height:3.009ex;" alt="{\displaystyle M|z|^{n}\leq |f(z)|}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\geq R,}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7ce311afc5cf9dfee94fdb8c1231502fb1001e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.891ex; height:2.843ex;" alt="{\displaystyle |z|\geq R,}" loading="lazy"> is necessarily a polynomial, of degree at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

</div>

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

Entire functions may grow as fast as any increasing function: for any increasing function

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:[0,\infty )\to [0,\infty )}</annotation>

</semantics>

</math><img src="./1c467e6d71d7f52d3b02317e77e79f90d4ffbc95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.811ex; height:2.843ex;" alt="{\displaystyle g:[0,\infty )\to [0,\infty )}" loading="lazy"> there exists an entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> such that

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)>g(|x|)}</annotation>

</semantics>

</math><img src="./d73018591e38cb3d18a7d7e2a4a788795bc244c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.065ex; height:2.843ex;" alt="{\displaystyle f(x)>g(|x|)}" loading="lazy"> for all real <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. Such a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> may be easily found of the form:

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=c+\sum _{k=1}^{\infty }\left({\frac {z}{k}}\right)^{n_{k}}}</annotation>

</semantics>

</math>

for a constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> and a strictly increasing sequence of positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b18f35b25d18a03f414bb9510599938e94c4768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.483ex; height:2.009ex;" alt="{\displaystyle n_{k}}" loading="lazy">. Any such sequence defines an entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy">, and if the powers are chosen appropriately we may satisfy the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)>g(|x|)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)>g(|x|)}</annotation>

</semantics>

</math><img src="./d73018591e38cb3d18a7d7e2a4a788795bc244c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.065ex; height:2.843ex;" alt="{\displaystyle f(x)>g(|x|)}" loading="lazy"> for all real <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. (For instance, it certainly holds if one chooses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c:=g(2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c:=g(2)}</annotation>

</semantics>

</math><img src="./bd579658d760b7aae81978bd7b4982e561d77c00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.84ex; height:2.843ex;" alt="{\displaystyle c:=g(2)}" loading="lazy"> and, for any integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d30d7dcf305b7bce39d36df72fe3985b47aa9961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 1}" loading="lazy"> one chooses an even exponent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b18f35b25d18a03f414bb9510599938e94c4768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.483ex; height:2.009ex;" alt="{\displaystyle n_{k}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {k+1}{k}}\right)^{n_{k}}\geq g(k+2)}">

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {k+1}{k}}\right)^{n_{k}}\geq g(k+2)}</annotation>

</semantics>

</math><img src="./d0639ecaf1a7a4a1dc75462a88f263b9aa373c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.787ex; height:6.176ex;" alt="{\displaystyle \left({\frac {k+1}{k}}\right)^{n_{k}}\geq g(k+2)}" loading="lazy">).

<div class="mw-heading mw-heading2"><h2 id="Order_and_type"> Order and type</h2></div>

The order (at infinity) of an entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> is defined using the <a href="Limit_superior" class="mw-redirect" title="Limit superior">limit superior</a> as:

</mrow>

</munder>

<mfrac>

<mrow>

<mrow>

<mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =\limsup _{r\to \infty }{\frac {\ln \left(\ln \|f\|_{\infty ,B_{r}}\right)}{\ln r}},}</annotation>

</semantics>

</math>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68e750df8ee313e34c90563be9f828d67c934b55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.509ex;" alt="{\displaystyle B_{r}}" loading="lazy"> is the disk of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{\infty ,B_{r}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|f\|_{\infty ,B_{r}}}</annotation>

</semantics>

</math><img src="./086c6016810367c490266f21341e12b4388e9bd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.95ex; height:3.009ex;" alt="{\displaystyle \|f\|_{\infty ,B_{r}}}" loading="lazy"> denotes the <a href="Supremum_norm" class="mw-redirect" title="Supremum norm">supremum norm</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68e750df8ee313e34c90563be9f828d67c934b55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.509ex;" alt="{\displaystyle B_{r}}" loading="lazy">. The order is a non-negative <a href="Real_number" title="Real number">real number</a> or infinity (except when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./083885a19c4077edc0bd0a6fd876b4326934de57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.437ex; height:2.843ex;" alt="{\displaystyle f(z)=0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">). In other words, the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> is the <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a> of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> such that:

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=O\left(\exp \left(|z|^{m}\right)\right),\quad {\text{as }}z\to \infty .}</annotation>

</semantics>

</math>

The example of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\exp(2z^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=\exp(2z^{2})}</annotation>

</semantics>

</math><img src="./c7239d00e41cd94a0d6c93d7bcd6202d4b3a7698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.943ex; height:3.176ex;" alt="{\displaystyle f(z)=\exp(2z^{2})}" loading="lazy"> shows that this does not mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=O(\exp(|z|^{m}))}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=O(\exp(|z|^{m}))}</annotation>

</semantics>

</math><img src="./a7175de811e5c3c4a8303a279961db2c2d67f39d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.276ex; height:3.009ex;" alt="{\displaystyle f(z)=O(\exp(|z|^{m}))}" loading="lazy"> if

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> is of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\rho <\infty ,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./464cd4e9b031027efa0b3034ab3eb21561991c7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.532ex; height:2.676ex;" alt="{\displaystyle 0<\rho <\infty ,}" loading="lazy"> one can also define the type:

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma =\limsup _{r\to \infty }{\frac {\ln \|f\|_{\infty ,B_{r}}}{r^{\rho }}}.}</annotation>

</semantics>

</math>

If the order is 1 and the type is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">, the function is said to be "of <a href="Exponential_type" title="Exponential type">exponential type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy">". If it is of order less than 1 it is said to be of exponential type 0.

If <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},}</annotation>

</semantics>

</math> then the order and type can be found by the formulas

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</munder>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{-\ln |a_{n}|}}\\[6pt](e\rho \sigma )^{\frac {1}{\rho }}&=\limsup _{n\to \infty }n^{\frac {1}{\rho }}|a_{n}|^{\frac {1}{n}}\end{aligned}}}</annotation>

</semantics>

</math>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(n)}}</annotation>

</semantics>

</math><img src="./9dfb1963ccde0e87eb3838f51dc19041e2ff3816.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.818ex; height:3.176ex;" alt="{\displaystyle f^{(n)}}" loading="lazy"> denote the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-th derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Then we may restate these formulas in terms of the derivatives at any arbitrary point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9e72d1d86e86355892b39b8eb32b964834e113bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{0}}" loading="lazy">:

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mtd>

<mtd>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

</mrow>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{n\ln n-\ln |f^{(n)}(z_{0})|}}=\left(1-\limsup _{n\to \infty }{\frac {\ln |f^{(n)}(z_{0})|}{n\ln n}}\right)^{-1}\\[6pt](\rho \sigma )^{\frac {1}{\rho }}&=e^{1-{\frac {1}{\rho }}}\limsup _{n\to \infty }{\frac {|f^{(n)}(z_{0})|^{\frac {1}{n}}}{n^{1-{\frac {1}{\rho }}}}}\end{aligned}}}</annotation>

</semantics>

</math>

The type may be infinite, as in the case of the <a href="Reciprocal_gamma_function" title="Reciprocal gamma function">reciprocal gamma function</a>, or zero (see example below under <a href="#Order_1">§ Order 1</a>).

Another way to find out the order and type is <a href="Matsaev's_theorem" title="Matsaev's theorem">Matsaev's theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

Here are some examples of functions of various orders:

<div class="mw-heading mw-heading4"><h4 id="Order_ρ">Order ρ</h4></div>

For arbitrary positive numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> one can construct an example of an entire function of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> and type <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> using:

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=\sum _{n=1}^{\infty }\left({\frac {e\rho \sigma }{n}}\right)^{\frac {n}{\rho }}z^{n}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Order_0">Order 0</h4></div>

<ul><li>Non-zero polynomials</li>

</mrow>

</mrow>

</munderover>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }2^{-n^{2}}z^{n}}</annotation>

</semantics>

</math><img src="./4a02924bdb952a5cdb96dc8fa96a33c6166dca13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.542ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }2^{-n^{2}}z^{n}}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Order_1/4">Order 1/4</h4></div>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\sqrt[{4}]{z}})}</annotation>

</semantics>

</math>

where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(u)=\cos(u)+\cosh(u)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u)=\cos(u)+\cosh(u)}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Order_1/3">Order 1/3</h4></div>

<mroot>

</mrow>

</mroot>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f({\sqrt[{3}]{z}})}</annotation>

</semantics>

</math>

where

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mfrac>

</mfrac>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mtext> a complex cube root of 1</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u)=e^{u}+e^{\omega u}+e^{\omega ^{2}u}=e^{u}+2e^{-{\frac {u}{2}}}\cos \left({\frac {{\sqrt {3}}u}{2}}\right),\quad {\text{with }}\omega {\text{ a complex cube root of 1}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Order_1/2">Order 1/2</h4></div>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \left(a{\sqrt {z}}\right)}</annotation>

</semantics>

</math> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f455a7f96d74aa94573d8e32da3b240ab0aa294f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" loading="lazy"> (for which the type is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =|a|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e4f526825b06add7fcc6728e22b25abc168a8ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.952ex; height:2.843ex;" alt="{\displaystyle \sigma =|a|}" loading="lazy">)

<div class="mw-heading mw-heading4"><h4 id="Order_1">Order 1</h4></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b74483ed4eafeaae141e5dbc58e6095c994270d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.68ex; height:2.843ex;" alt="{\displaystyle \exp(az)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f455a7f96d74aa94573d8e32da3b240ab0aa294f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =|a|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e4f526825b06add7fcc6728e22b25abc168a8ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.952ex; height:2.843ex;" alt="{\displaystyle \sigma =|a|}" loading="lazy">)</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./292b02242178488542ee5c526562b6f1cad812c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.753ex; height:2.843ex;" alt="{\displaystyle \sin(z)}" loading="lazy"></li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2075ae5e2034bd0954ac61f6cc5bf2304b067dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.301ex; height:2.843ex;" alt="{\displaystyle \cosh(z)}" loading="lazy"></li>

<li>the <a href="Bessel_function" title="Bessel function">Bessel functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b084e7a8cc974442131c4b789255bf7b01242e18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.406ex; height:2.843ex;" alt="{\displaystyle J_{n}(z)}" loading="lazy"> and spherical Bessel functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{n}(z)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./80916039ca927f44ce0f98ac7144e0ee91ee2347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:5.101ex; height:2.843ex;" alt="{\displaystyle j_{n}(z)}" loading="lazy"> for integer values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"><a href="#cite_note-7">[2]</a></li>

<li>the <a href="Reciprocal_gamma_function" title="Reciprocal gamma function">reciprocal gamma function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\Gamma (z)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1d0172e7e34787959b2c38ee6412d6a051d8519c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.675ex; height:2.843ex;" alt="{\displaystyle 1/\Gamma (z)}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is infinite)</li>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{\infty }{\frac {z^{n}}{(n\ln n)^{n}}}.\quad (\sigma =0)}</annotation>

</semantics>

</math><img src="./9d82b7e990a1d55c92388b62911f315f194b56d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.865ex; height:6.843ex;" alt="{\displaystyle \sum _{n=2}^{\infty }{\frac {z^{n}}{(n\ln n)^{n}}}.\quad (\sigma =0)}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Order_3/2">Order 3/2</h4></div>

<ul><li><a href="Airy_function" title="Airy function">Airy function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ai(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3efe9bd6e5db9ef2253f5d6eb33c1ae0823340fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.443ex; height:2.843ex;" alt="{\displaystyle Ai(z)}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Order_2">Order 2</h4></div>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(az^{2})}</annotation>

</semantics>

</math><img src="./db727750e6a9ae6bfebc28d6544e2b5e07fd1b73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.736ex; height:3.176ex;" alt="{\displaystyle \exp(az^{2})}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f455a7f96d74aa94573d8e32da3b240ab0aa294f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =|a|}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3e4f526825b06add7fcc6728e22b25abc168a8ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.952ex; height:2.843ex;" alt="{\displaystyle \sigma =|a|}" loading="lazy">)</li>

<li>The <a href="Barnes_G-function" title="Barnes G-function">Barnes G-function</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is infinite).</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Order_infinity">Order infinity</h4></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fb2022c250e84276d5906ce1fb6c9b97fd87d61c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.812ex; height:2.843ex;" alt="{\displaystyle \exp(\exp(z))}" loading="lazy"></li></ul>

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

Entire functions of finite order have <a href="Jacques_Hadamard" title="Jacques Hadamard">Hadamard</a>'s canonical representation (<a href="Hadamard_factorization_theorem" title="Hadamard factorization theorem">Hadamard factorization theorem</a>):

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)=z^{m}e^{P(z)}\prod _{n=1}^{\infty }\left(1-{\frac {z}{z_{n}}}\right)\exp \left({\frac {z}{z_{n}}}+\cdots +{\frac {1}{p}}\left({\frac {z}{z_{n}}}\right)^{p}\right),}</annotation>

</semantics>

</math>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a51cdfac24f8b95ed711f11ea9502da4087b6a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.17ex; height:2.009ex;" alt="{\displaystyle z_{k}}" loading="lazy"> are those <a href="Zero_of_a_function" title="Zero of a function">roots</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> that are not zero (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}\neq 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b16f35ec2f2a16693924ca50e055c557f2c723d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.431ex; height:2.676ex;" alt="{\displaystyle z_{k}\neq 0}" loading="lazy">), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the order of the zero of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b92bfc06485cc90286474b14a516a68d8bfdd7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}" loading="lazy"> (the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e57f21007575fd03e3be0da20af34d25829cc9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}" loading="lazy"> being taken to mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./370f1a75139206568bf528e86092088c4883d8c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)\neq 0}" loading="lazy">), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> a polynomial (whose degree we shall call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is the smallest non-negative integer such that the series

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{|z_{n}|^{p+1}}}}</annotation>

</semantics>

</math>

converges. The non-negative integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\max\{p,q\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=\max\{p,q\}}</annotation>

</semantics>

</math><img src="./d4ae9b08cebc72d412e47591bfa47286cb8228e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.138ex; height:2.843ex;" alt="{\displaystyle g=\max\{p,q\}}" loading="lazy"> is called the genus of the entire function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

If the order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is not an integer, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=[\rho ]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=[\rho ]}</annotation>

</semantics>

</math><img src="./0daa8fd34b7cfa4ebcc0b39ec866af4d4a76a48d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle g=[\rho ]}" loading="lazy"> is the integer part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">. If the order is a positive integer, then there are two possibilities: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\rho -1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0065630cf688bb287b3b16e8d670c1c3cca07707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.419ex; height:2.676ex;" alt="{\displaystyle g=\rho -1}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a3647533e4affb4ac9330b35d22738afba9abcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.416ex; height:2.176ex;" alt="{\displaystyle g=\rho }" loading="lazy">.

For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ee55beec18afd710e7ab767964b915b020c65093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e473a3de151d75296f141f9f482fe59d582a7509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d1185b570f67b4221307626254f64f9e619e769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.552ex; height:2.009ex;" alt="{\displaystyle \exp }" loading="lazy"> are entire functions of genus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\rho =1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./70a15427cc9693d9fc8a1bab19f04b9de875b46c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.677ex; height:2.676ex;" alt="{\displaystyle g=\rho =1}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Other_examples">Other examples</h2></div>

According to <a href="J._E._Littlewood" class="mw-redirect" title="J. E. Littlewood">J. E. Littlewood</a>, the <a href="Weierstrass_sigma_function" class="mw-redirect" title="Weierstrass sigma function">Weierstrass sigma function</a> is a 'typical' entire function. This statement can be made precise in the theory of random entire functions: the asymptotic behavior of almost all entire functions is similar to that of the sigma function. Other examples include the <a href="Fresnel_integral" title="Fresnel integral">Fresnel integrals</a>, the <a href="Jacobi_theta_function" class="mw-redirect" title="Jacobi theta function">Jacobi theta function</a>, and the <a href="Reciprocal_Gamma_function" class="mw-redirect" title="Reciprocal Gamma function">reciprocal Gamma function</a>. The exponential function and the error function are special cases of the <a href="Mittag-Leffler_function" title="Mittag-Leffler function">Mittag-Leffler function</a>. According to the fundamental <a href="Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">theorem of Paley and Wiener</a>, <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> of functions (or distributions) with bounded support are entire functions of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> and finite type.

Other examples are solutions of linear differential equations with polynomial coefficients. If the coefficient at the highest derivative is constant, then all solutions of such equations are entire functions. For example, the exponential function, sine, cosine, <a href="Airy_function" title="Airy function">Airy functions</a> and <a href="Parabolic_cylinder_function" title="Parabolic cylinder function">Parabolic cylinder functions</a> arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to study <a href="Holomorphic_dynamics" class="mw-redirect" title="Holomorphic dynamics">dynamics of entire functions</a>.

An entire function of the square root of a complex number is entire if the original function is <a href="Even_function" class="mw-redirect" title="Even function">even</a>, for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos({\sqrt {z}})}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a3179bac3484fbb7636bd57024cddddcacd36ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.944ex; height:3.009ex;" alt="{\displaystyle \cos({\sqrt {z}})}" loading="lazy">.

If a sequence of polynomials all of whose roots are real converges in a neighborhood of the origin to a limit which is not identically equal to zero, then this limit is an entire function. Such entire functions form the <a href="Laguerre%E2%80%93P%C3%B3lya_class" title="Laguerre–Pólya class">Laguerre–Pólya class</a>, which can also be characterized in terms of the Hadamard product, namely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> belongs to this class <a href="If_and_only_if" title="If and only if">if and only if</a> in the Hadamard representation all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5e8228e40cfaa6cb1f163c066ae7054faeff8c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle z_{n}}" loading="lazy"> are real, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \leq 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a7272729c5ddc8692703612118df631b03035d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho \leq 1}" loading="lazy">, and

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(z)=a+bz+cz^{2}}</annotation>

</semantics>

</math><img src="./3cba82b9338e06e6d726c2372438c3c0562ca891.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.889ex; height:3.176ex;" alt="{\displaystyle P(z)=a+bz+cz^{2}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> are real, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\leq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./13bf66eac84c146d83f1a8cc261bf10d09e118fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.268ex; height:2.343ex;" alt="{\displaystyle c\leq 0}" loading="lazy">. For example, the sequence of polynomials

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-{\frac {(z-d)^{2}}{n}}\right)^{n}}</annotation>

</semantics>

</math>

converges, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> increases, to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(-(z-d)^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(-(z-d)^{2})}</annotation>

</semantics>

</math><img src="./43658a6469eea26f9a02f6db89faef7375dec17d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.178ex; height:3.176ex;" alt="{\displaystyle \exp(-(z-d)^{2})}" loading="lazy">. The polynomials

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\left(\left(1+{\frac {iz}{n}}\right)^{n}+\left(1-{\frac {iz}{n}}\right)^{n}\right)}</annotation>

</semantics>

</math>

have all real roots, and converge to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./48e7d26bd35f069db0efa8925a0f93befc6d0566.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.009ex; height:2.843ex;" alt="{\displaystyle \cos(z)}" loading="lazy">. The polynomials

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{m=1}^{n}\left(1-{\frac {z^{2}}{\left(\left(m-{\frac {1}{2}}\right)\pi \right)^{2}}}\right)}</annotation>

</semantics>

</math>

also converge to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./48e7d26bd35f069db0efa8925a0f93befc6d0566.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.009ex; height:2.843ex;" alt="{\displaystyle \cos(z)}" loading="lazy">, showing the buildup of the Hadamard product for cosine.

<ul><li><a href="Jensen's_formula" title="Jensen's formula">Jensen's formula</a></li>

<li><a href="Carlson's_theorem" title="Carlson's theorem">Carlson's theorem</a></li>

<li><a href="Exponential_type" title="Exponential type">Exponential type</a></li>

<li><a href="Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorem</a></li>

<li><a href="Wiman-Valiron_theory" title="Wiman-Valiron theory">Wiman-Valiron theory</a></li></ul>

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

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-1"><a href="#cite_ref-1">^</a> If necessary, the logarithm of zero is taken to be equal to minus infinity.

</li>

For instance, if the real part is known on part of the unit circle, then it is known on the whole unit circle by <a href="Analytic_extension" class="mw-redirect" title="Analytic extension">analytic extension</a>, and then the coefficients of the infinite series are determined from the coefficients of the <a href="Fourier_series" title="Fourier series">Fourier series</a> for the real part on the unit circle.

</li>

Liouville's theorem may be used to elegantly prove the <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>.

</li>

The <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a> is the whole complex plane augmented with a single point at infinity.

</li>

The converse is also true as for any polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(z)=\sum _{k=0}^{n}a_{k}z^{k}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle p(z)=\sum _{k=0}^{n}a_{k}z^{k}}</annotation>

</semantics>

</math><img src="./4dac88e6f775943148c7cbd391cd4b734ea90824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:17.783ex; height:3.176ex;" alt="{\textstyle p(z)=\sum _{k=0}^{n}a_{k}z^{k}}" loading="lazy"> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |p(z)|\leq \left(\ \sum _{k=0}^{n}|a_{k}|\ \right)|z|^{n}}">

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |p(z)|\leq \left(\ \sum _{k=0}^{n}|a_{k}|\ \right)|z|^{n}}</annotation>

</semantics>

</math><img src="./085f1441c6f73d93875e6114cb1e4fe4dcde5d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.446ex; height:3.176ex;" alt="{\textstyle |p(z)|\leq \left(\ \sum _{k=0}^{n}|a_{k}|\ \right)|z|^{n}}" loading="lazy"> holds for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|\geq 1~.}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d819c3d0c0ebca63536b78bc4c317bd945f247e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.87ex; height:2.843ex;" alt="{\displaystyle |z|\geq 1~.}" loading="lazy">

</li>

</ol></div></div>

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

<li id="cite_note-FOOTNOTEBoas19541-2"><a href="#cite_ref-FOOTNOTEBoas19541_2-0">^</a> <a href="#CITEREFBoas1954">Boas 1954</a>, p. 1.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> See asymptotic expansion in Abramowitz and Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/page_377.htm">p. 377, 9.7.1</a>.

</li>

</ol></div>

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

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</style><cite id="CITEREFBoas1954" class="citation book cs1"><a href="Ralph_P._Boas%2C_Jr." class="mw-redirect" title="Ralph P. Boas, Jr.">Boas, Ralph P.</a> (1954). Entire Functions. <a href="Academic_Press" title="Academic Press">Academic Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080873138</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/847696">847696</a>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li>

<li><cite id="CITEREFLevin1980" class="citation book cs1">Levin, B. Ya. (1980) [1964]. Distribution of Zeros of Entire Functions. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4505-9</bdi>.</cite></li>

<li><cite id="CITEREFLevin1996" class="citation book cs1">Levin, B. Ya. (1996). Lectures on Entire Functions. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-0897-9</bdi>.</cite></li></ul>

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