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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For the use of the term in calculus, see <a href="Linear_function_(calculus)" title="Linear function (calculus)">Linear function (calculus)</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the term <b>linear function</b> refers to two distinct but related notions:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<ul><li>In <a href="Calculus" title="Calculus">calculus</a> and related areas, a linear function is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> whose <a href="Graph_of_a_function" title="Graph of a function">graph</a> is a <a href="Straight_line" class="mw-redirect" title="Straight line">straight line</a>, that is, a <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> of <a href="Polynomial_degree" class="mw-redirect" title="Polynomial degree">degree</a> zero or one.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For distinguishing such a linear function from the other concept, the term <i><a href="Affine_function" class="mw-redirect" title="Affine function">affine function</a></i> is often used.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li>In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, a linear function is a <a href="Linear_map" title="Linear map">linear map</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="As_a_polynomial_function">As a polynomial function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_function_(calculus)" title="Linear function (calculus)">Linear function (calculus)</a></div>

<p>In calculus, <a href="Analytic_geometry" title="Analytic geometry">analytic geometry</a> and related areas, a linear function is a polynomial of degree one or less, including the <a href="Zero_polynomial" class="mw-redirect" title="Zero polynomial">zero polynomial</a> (the latter not being considered to have degree zero).

</p><p>When the function is of only one <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a>, it is of the form

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax+b,}</annotation>

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</math></span><img src="./4bb78c557e6ea8b777bb975d3ef12ab17bb411cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.56ex; height:2.843ex;" alt="{\displaystyle f(x)=ax+b,}" loading="lazy"></span></dd></dl>

<p>where <span class="texhtml mvar" style="font-style:italic;"><i>a</i></span> and <span class="texhtml mvar" style="font-style:italic;"><i>b</i></span> are <a href="Constant_(mathematics)" title="Constant (mathematics)">constants</a>, often <a href="Real_number" title="Real number">real numbers</a>. The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of such a function of one variable is a nonvertical line. <span class="texhtml mvar" style="font-style:italic;"><i>a</i></span> is frequently referred to as the slope of the line, and <span class="texhtml mvar" style="font-style:italic;"><i>b</i></span> as the intercept.

</p><p>If <i>a > 0</i> then the <a href="Slope" title="Slope">gradient</a> is positive and the graph slopes upwards.

</p><p>If <i>a < 0</i> then the <a href="Slope" title="Slope">gradient</a> is negative and the graph slopes downwards.

</p><p>For a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1},\ldots ,x_{k})}">

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<annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{k})}</annotation>

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</math></span><img src="./7afedaaa7dfef56c4f5825bc313635b1c6755af3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.068ex; height:2.843ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{k})}" loading="lazy"></span> of any finite number of variables, the general formula is

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<annotation encoding="application/x-tex">{\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},}</annotation>

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</math></span><img src="./eb5a25377a0e452b240c75d17695d347a33e0e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.461ex; height:2.843ex;" alt="{\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},}" loading="lazy"></span></dd></dl>

<p>and the graph is a <a href="Hyperplane" title="Hyperplane">hyperplane</a> of dimension <span class="nowrap"><i>k</i></span>.

</p><p>A <a href="Constant_function" title="Constant function">constant function</a> is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.

</p><p>In this context, a function that is also a linear map (the other meaning) may be referred to as a <a href="Homogeneous_function" title="Homogeneous function">homogeneous</a> linear function or a <a href="Linear_form" title="Linear form">linear form</a>. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued <a href="Affine_map" class="mw-redirect" title="Affine map">affine maps</a>.

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<div class="mw-heading mw-heading2"><h2 id="As_a_linear_map">As a linear map</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_map" title="Linear map">Linear map</a></div>

<p>In linear algebra, a linear function is a map <i>f</i> between two <a href="Vector_space" title="Vector space">vector spaces</a> such that

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

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</math></span><img src="./20c1db57416b4644aacdfc90beb5481f08e39462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.687ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle f(a\mathbf {x} )=af(\mathbf {x} ).}</annotation>

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</math></span><img src="./79be6bd64f9093829d58fc9e67ee5529eb0a848d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.203ex; height:2.843ex;" alt="{\displaystyle f(a\mathbf {x} )=af(\mathbf {x} ).}" loading="lazy"></span></dd></dl>

<p>Here <span class="texhtml"><i>a</i></span> denotes a constant belonging to some <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml"><i>K</i></span> of <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> (for example, the <a href="Real_number" title="Real number">real numbers</a>) and <span class="texhtml"><b>x</b></span> and <span class="texhtml"><b>y</b></span> are elements of a <a href="Vector_space" title="Vector space">vector space</a>, which might be <span class="texhtml"><i>K</i></span> itself.

</p><p>In other terms the linear function preserves <a href="Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a> and <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>.

</p><p>Some authors use "linear function" only for linear maps that take values in the scalar field;<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> these are more commonly called <a href="Linear_form" title="Linear form">linear forms</a>.

</p><p>The "linear functions" of calculus qualify as "linear maps" when (and only when) <span class="texhtml"><i>f</i>(0, ..., 0) = 0</span>, or, equivalently, when the constant <span class="texhtml mvar" style="font-style:italic;">b</span> equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

</p>

<ul><li><a href="Homogeneous_function" title="Homogeneous function">Homogeneous function</a></li>

<li><a href="Nonlinear_system" title="Nonlinear system">Nonlinear system</a></li>

<li><a href="Piecewise_linear_function" title="Piecewise linear function">Piecewise linear function</a></li>

<li><a href="Linear_approximation" title="Linear approximation">Linear approximation</a></li>

<li><a href="Linear_interpolation" title="Linear interpolation">Linear interpolation</a></li>

<li><a href="Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous linear map</a></li>

<li><a href="Linear_least_squares" title="Linear least squares">Linear least squares</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">"The term <i>linear function</i> means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Stewart 2012, p. 23</span>

</li>

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</style><cite id="CITEREFA._Kurosh1975" class="citation book cs1">A. Kurosh (1975). <i>Higher Algebra</i>. Mir Publishers. p. 214.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFT._M._Apostol1981" class="citation book cs1">T. M. Apostol (1981). <i>Mathematical Analysis</i>. Addison-Wesley. p. 345.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Shores 2007, p. 71</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Gelfand 1961</span>

</li>

</ol></div>

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

<ul><li>Izrail Moiseevich Gelfand (1961), <i>Lectures on Linear Algebra</i>, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-66082-6</bdi></li>

<li><cite id="CITEREFShores2007" class="citation book cs1">Shores, Thomas S. (2007). <i>Applied Linear Algebra and Matrix Analysis</i>. <a href="Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-33195-9</bdi>.</cite></li>

<li><cite id="CITEREFStewart2012" class="citation book cs1">Stewart, James (2012). <i>Calculus: Early Transcendentals</i> (7E ed.). Brooks/Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-538-49790-9</bdi>.</cite></li>

<li>Leonid N. Vaserstein (2006), "Linear Programming", in <a href="Leslie_Hogben" title="Leslie Hogben">Leslie Hogben</a>, ed., <i>Handbook of Linear Algebra</i>, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-584-88510-6</bdi></li></ul>

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</style><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li>

<li><a href="Concave_function" title="Concave function">Concave function</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous function</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Finite_difference" title="Finite difference">Finite difference</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li>

<li><a href="Graph_of_a_function" title="Graph of a function">Graph of a function</a></li>

<li><a href="Radian" title="Radian">Radian</a></li>

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

<li><a href="Secant_line" title="Secant line">Secant</a></li>

<li><a href="Slope" title="Slope">Slope</a></li>

<li><a href="Tangent" title="Tangent">Tangent</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li>

<li><a href="Limit_of_a_function" title="Limit of a function">Limit of a function</a>

<ul><li><a href="One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li>

<li><a href="Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li>

<li><a href="Order_of_approximation" title="Order of approximation">Order of approximation</a></li>

<li><a href="(%CE%B5%2C_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Derivative" title="Derivative">Derivative</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li>

<li><a href="Differential_operator" title="Differential operator">Differential operator</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Notation_for_differentiation" title="Notation for differentiation">Notation</a>

<ul><li><a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a></li>

<li><a href="Newton's_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li>

<li><a href="Differentiation_rules" title="Differentiation rules">Rules of differentiation</a>

<ul><li><a href="Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's</a></li>

<li><a href="Product_rule" title="Product rule">Product</a>

<ul><li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li>

<li>Other techniques

<ul><li><a href="Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li>

<li><a href="Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li>

<li><a href="Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li></ul></li>

<li><a href="Stationary_point" title="Stationary point">Stationary points</a>

<ul><li><a href="First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li>

<li><a href="Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li>

<li><a href="Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li>

<li><a href="Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li>

<li>Further applications

<ul><li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

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

<li><a href="Differential_equation" title="Differential equation">Differential equation</a>

<ul><li><a href="Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li>

<li><a href="Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Arc_length" title="Arc length">Arc length</a></li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Integral#Properties" title="Integral">Basic properties</a></li>

<li><a href="Constant_of_integration" title="Constant of integration">Constant of integration</a></li>

<li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>

<ul><li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li>

<li><a href="Integration_by_parts" title="Integration by parts">Integration by parts</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Integration by substitution</a>

<ul><li><a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li>

<li><a href="Euler_substitution" title="Euler substitution">Euler</a></li>

<li><a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a>

<ul><li><a href="Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li>

<li><a href="Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li>

<li>Volumes

<ul><li><a href="Disc_integration" title="Disc integration">Washer method</a></li>

<li><a href="Shell_integration" title="Shell integration">Shell method</a></li></ul></li>

<li><a href="Integral_equation" title="Integral equation">Integral equation</a></li>

<li><a href="Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Derivatives

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li>

<li>Basic theorems

<ul><li><a href="Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li>

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

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li>Advanced topics

<ul><li><a href="Differential_form" title="Differential form">Differential forms</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Generalized_Stokes'_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li>

<li>Types of series

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier</a></li>

<li><a href="Geometric_series" title="Geometric series">Geometric</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li>

<li><a href="Power_series" title="Power series">Power</a>

<ul><li><a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li>

<li>Tests of convergence

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet's</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br>and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li>

<li><a href="E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li>

<li><a href="Exponential_function" title="Exponential function">Exponential function</a></li>

<li><a href="Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li>

<li><a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adequality" title="Adequality">Adequality</a></li>

<li><a href="Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li>

<li><a href="Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li>

<li><a href="Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li>

<li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Infinitesimal" title="Infinitesimal">Infinitesimal</a></li>

<li><a href="Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li>

<li><a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li>

<li><a href="Fluxion" title="Fluxion">Fluxion</a></li>

<li><a href="Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><i><a href="Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li>

<li><i><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li>

<li><a href="List_of_integrals_of_irrational_algebraic_functions" title="List of integrals of irrational algebraic functions">irrational algebraic functions</a></li>

<li><a href="List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li>

<li><a href="List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li>

<li><a href="List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a>

<ul><li><a href="List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li>

<li><a href="List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a>

<ul><li><a href="List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li>

<li><a href="Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li>

<li><a href="Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_limits" title="List of limits">List of limits</a></li>

<li><a href="Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Complex calculus

<ul><li><a href="Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li>

<li>Differential geometry

<ul><li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Curvature" title="Curvature">Curvature</a></li>

<li><a href="Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li>

<li><a href="Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li></ul></li>

<li><a href="Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li>

<li><a href="Gabriel's_horn" title="Gabriel's horn">Gabriel's horn</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li>

<li><a href="Regiomontanus'_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li>

<li><a href="Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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