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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_by_analytic_majorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proof by analytic majorization</span> </div> </a> <ul id="toc-Proof_by_analytic_majorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-order_Cauchy–Kovalevskaya_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_Cauchy–Kovalevskaya_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Higher-order Cauchy–Kovalevskaya theorem</span> </div> </a> <button aria-controls="toc-Higher-order_Cauchy–Kovalevskaya_theorem-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Higher-order Cauchy–Kovalevskaya theorem subsection</span> </button> <ul id="toc-Higher-order_Cauchy–Kovalevskaya_theorem-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cauchy–Kovalevskaya–Kashiwara_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cauchy–Kovalevskaya–Kashiwara_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Cauchy–Kovalevskaya–Kashiwara theorem</span> </div> </a> <button aria-controls="toc-Cauchy–Kovalevskaya–Kashiwara_theorem-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Cauchy–Kovalevskaya–Kashiwara theorem 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class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Scope</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;"> <div class="hlist"><ul><li><a href="/wiki/Natural_science" title="Natural science">Natural sciences</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Physics" title="Physics">Physics</a></li> <li><a href="/wiki/Chemistry" title="Chemistry">Chemistry</a></li> <li><br /><a href="/wiki/Biology" title="Biology">Biology</a></li> <li><a href="/wiki/Geology" title="Geology">Geology</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;"> <ul><li><a href="/wiki/Economics" title="Economics">Economics</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td> </tr></tbody></table> <hr /> <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;"> Classification</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> By variable type</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li> <li>Coupled / Decoupled</li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Features</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Operator</a></li></ul> </div> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size:85%;">(discrete analogue)</span></a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Solution</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem </a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Initial_condition" title="Initial condition">Initial conditions</a></li> <li><a href="/wiki/Boundary_value_problem" title="Boundary value problem">Boundary values</a> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><span class="nowrap"><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li>Inspection</li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li> <li><br /><a href="/wiki/Euler_method" title="Euler method">Euler</a></li> <li><a href="/wiki/Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size:85%;">(<a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element</a> <ul><li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li> <li><a href="/wiki/Green%27s_function" title="Green's function">Green's function</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul> </div> <ul><li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> People</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/George_Green_(mathematician)" title="George Green (mathematician)">George Green</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations" title="Template:Differential equations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations" title="Template talk:Differential equations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations" title="Special:EditPage/Template:Differential equations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Cauchy–Kovalevskaya theorem</b> (also written as the <b>Cauchy–Kowalevski theorem</b>) is the main local <a href="/wiki/Existence_theorem" title="Existence theorem">existence</a> and uniqueness theorem for <a href="/wiki/Analytic_function" title="Analytic function">analytic</a> <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> associated with <a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy initial value problems</a>. A special case was proven by <a href="/wiki/Augustin_Cauchy" class="mw-redirect" title="Augustin Cauchy">Augustin Cauchy</a> (<a href="#CITEREFCauchy1842">1842</a>), and the full result by <a href="/wiki/Sofia_Kovalevskaya" class="mw-redirect" title="Sofia Kovalevskaya">Sofya Kovalevskaya</a> (<a href="#CITEREFKovalevskaya1874">1874</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="First_order_Cauchy–Kovalevskaya_theorem"><span id="First_order_Cauchy.E2.80.93Kovalevskaya_theorem"></span>First order Cauchy–Kovalevskaya theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=1" title="Edit section: First order Cauchy–Kovalevskaya theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This theorem is about the existence of solutions to a system of <i>m</i> differential equations in <i>n</i> dimensions when the coefficients are <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>. The theorem and its proof are valid for analytic functions of either real or complex variables. </p><p>Let <i>K</i> denote either the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> of real or complex numbers, and let <i>V</i> = <i>K</i><sup><i>m</i></sup> and <i>W</i> = <i>K</i><sup><i>n</i></sup>. Let <i>A</i><sub>1</sub>, ..., <i>A</i><sub><i>n</i>−1</sub> be <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> defined on some <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of (0, 0) in <i>W</i> × <i>V</i> and taking values in the <i>m</i> × <i>m</i> matrices, and let <i>b</i> be an analytic function with values in <i>V</i> defined on the same neighbourhood. Then there is a neighbourhood of 0 in <i>W</i> on which the <a href="/wiki/Partial_differential_equation#Linear_and_nonlinear_equations" title="Partial differential equation">quasilinear</a> <a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45676c70a499b430aee5015405602f373ba1561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.078ex; height:3.009ex;" alt="{\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}"></span></dd></dl> <p>with initial condition </p> <dl><dd><span class="mwe-math-element"><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)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}"></span></dd></dl> <p>on the hypersurface </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfea86310aa3db4720d6d0ab82a1016c749c97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.809ex; height:2.509ex;" alt="{\displaystyle x_{n}=0}"></span></dd></dl> <p>has a unique analytic solution <i>ƒ</i> : <i>W</i> → <i>V</i> near 0. </p><p><a href="/wiki/Lewy%27s_example" title="Lewy's example">Lewy's example</a> shows that the theorem is not more generally valid for all smooth functions. </p><p>The theorem can also be stated in abstract (real or complex) vector spaces. Let <i>V</i> and <i>W</i> be finite-dimensional real or complex vector spaces, with <i>n</i> = dim <i>W</i>. Let <i>A</i><sub>1</sub>, ..., <i>A</i><sub><i>n</i>−1</sub> be <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> with values in <a href="/wiki/Endomorphism" title="Endomorphism">End (<i>V</i>)</a> and <i>b</i> an analytic function with values in <i>V</i>, defined on some <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of (0, 0) in <i>W</i> × <i>V</i>. In this case, the same result holds. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_by_analytic_majorization">Proof by analytic majorization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=2" title="Edit section: Proof by analytic majorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Both sides of the <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> can be expanded as <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> and give recurrence relations for the coefficients of the formal power series for <i>f</i> that uniquely determine the coefficients. The <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> coefficients of the <i>A</i><sub><i>i</i></sub>'s and <i>b</i> are <a href="/wiki/Majorization" title="Majorization">majorized</a> in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the <i>A</i><sub><i>i</i></sub>'s and <i>b</i> has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge where the scalar solution converges. </p> <div class="mw-heading mw-heading2"><h2 id="Higher-order_Cauchy–Kovalevskaya_theorem"><span id="Higher-order_Cauchy.E2.80.93Kovalevskaya_theorem"></span>Higher-order Cauchy–Kovalevskaya theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=3" title="Edit section: Higher-order Cauchy–Kovalevskaya theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>F</i> and <i>f</i><sub><i>j</i></sub> are analytic functions near 0, then the <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> Cauchy problem </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}^{k}h=F\left(x,t,\partial _{t}^{j}\,\partial _{x}^{\alpha }h\right),{\text{ where }}j<k{\text{ and }}|\alpha |+j\leq k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mi>h</mi> <mo>=</mo> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> where </mtext> </mrow> <mi>j</mi> <mo><</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>j</mi> <mo>≤</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}^{k}h=F\left(x,t,\partial _{t}^{j}\,\partial _{x}^{\alpha }h\right),{\text{ where }}j<k{\text{ and }}|\alpha |+j\leq k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75123b2049704891c0c95ffc5ce8b30f1097820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.859ex; height:4.843ex;" alt="{\displaystyle \partial _{t}^{k}h=F\left(x,t,\partial _{t}^{j}\,\partial _{x}^{\alpha }h\right),{\text{ where }}j<k{\text{ and }}|\alpha |+j\leq k,}"></span></dd></dl> <p>with initial conditions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo><</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1563a4534688d469049b8c1fe80b948296bda365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.068ex; height:3.509ex;" alt="{\displaystyle \partial _{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,}"></span></dd></dl> <p>has a unique analytic solution near 0. </p><p>This follows from the first order problem by considering the derivatives of <i>h</i> appearing on the right hand side as components of a vector-valued function. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=4" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}h=\partial _{x}^{2}h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>h</mi> <mo>=</mo> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}h=\partial _{x}^{2}h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/375becb4026e26b41495ce510d304732fe81bf66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.244ex; height:2.843ex;" alt="{\displaystyle \partial _{t}h=\partial _{x}^{2}h}"></span></dd></dl> <p>with the condition </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(0,x)={1 \over 1+x^{2}}{\text{ for }}t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(0,x)={1 \over 1+x^{2}}{\text{ for }}t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d773319e0443b902018d5bdada66e105c591ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.043ex; height:5.676ex;" alt="{\displaystyle h(0,x)={1 \over 1+x^{2}}{\text{ for }}t=0}"></span></dd></dl> <p>has a unique formal power series solution (expanded around (0, 0)). However this formal power series does not converge for any non-zero values of <i>t</i>, so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |<i>α</i>| + <i>j</i> ≤ <i>k</i> above cannot be dropped. (This example is due to Kowalevski.) </p> <div class="mw-heading mw-heading2"><h2 id="Cauchy–Kovalevskaya–Kashiwara_theorem"><span id="Cauchy.E2.80.93Kovalevskaya.E2.80.93Kashiwara_theorem"></span>Cauchy–Kovalevskaya–Kashiwara theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=5" title="Edit section: Cauchy–Kovalevskaya–Kashiwara theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the <a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya%E2%80%93Kashiwara_theorem&action=edit&redlink=1" class="new" title="Cauchy–Kovalevskaya–Kashiwara theorem (page does not exist)">Cauchy–Kovalevskaya–Kashiwara theorem</a>, due to <a href="/wiki/Masaki_Kashiwara" title="Masaki Kashiwara">Masaki Kashiwara</a> (<a href="#CITEREFKashiwara1983">1983</a>). This theorem involves a <a href="/wiki/Cohomological" class="mw-redirect" title="Cohomological">cohomological</a> formulation, presented in the language of <a href="/wiki/D-modules" class="mw-redirect" title="D-modules">D-modules</a>. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a <a href="/wiki/Derived_functor" title="Derived functor">derived functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ext^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>x</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ext^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f95bd54a6abfe9a2efb90b0b8ff3d88df50f8b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.999ex; height:2.676ex;" alt="{\displaystyle Ext^{1}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=6" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1be80a749be067190db51891139a12d4886fc054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.534ex; height:2.176ex;" alt="{\displaystyle n\leq m}"></span>. Set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\{x_{1}=\cdots =x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=\{x_{1}=\cdots =x_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24674fc22b1ec33500432a646bfe408965a5d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.049ex; height:2.843ex;" alt="{\displaystyle Y=\{x_{1}=\cdots =x_{n}\}}"></span>. The system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{x_{i}}f=g_{i},i=1,\ldots ,n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mi>f</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{x_{i}}f=g_{i},i=1,\ldots ,n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477b945db65fb6030c6a8df41c3584bcae9e9e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.635ex; height:2.843ex;" alt="{\displaystyle \partial _{x_{i}}f=g_{i},i=1,\ldots ,n,}"></span> has a solution <span class="mwe-math-element"><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\in \mathbb {C} \{x_{1},\ldots ,x_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {C} \{x_{1},\ldots ,x_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f75f42e837131bc1d4931c4447fce855ac8f3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.689ex; height:2.843ex;" alt="{\displaystyle f\in \mathbb {C} \{x_{1},\ldots ,x_{m}\}}"></span> if and only if the compatibility conditions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{x_{i}}g_{j}=\partial _{x_{j}}g_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{x_{i}}g_{j}=\partial _{x_{j}}g_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9c2896899a93bdab83258d75a8d5fa6826028e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.179ex; height:3.009ex;" alt="{\displaystyle \partial _{x_{i}}g_{j}=\partial _{x_{j}}g_{i}}"></span> are verified. In order to have a unique solution we must include an initial condition <span class="mwe-math-element"><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|_{Y}=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f|_{Y}=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52fdeab09695124b66260bc88e808d8b67eaf95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.849ex; height:2.843ex;" alt="{\displaystyle f|_{Y}=h}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in \mathbb {C} \{x_{n+1},\ldots ,x_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in \mathbb {C} \{x_{n+1},\ldots ,x_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b1cfe840c8fe948feb2c967ddfbb20e6accf23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.014ex; height:2.843ex;" alt="{\displaystyle h\in \mathbb {C} \{x_{n+1},\ldots ,x_{m}\}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCauchy1842" class="citation cs2">Cauchy, Augustin (1842), <a rel="nofollow" class="external text" href="https://archive.org/stream/oeuvresdaugusti107caucrich#page/n27/mode/2up">"Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles"</a>, <i>Comptes rendus</i>, <b>15</b></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comptes+rendus&rft.atitle=M%C3%A9moire+sur+l%27emploi+du+calcul+des+limites+dans+l%27int%C3%A9gration+des+%C3%A9quations+aux+d%C3%A9riv%C3%A9es+partielles&rft.volume=15&rft.date=1842&rft.aulast=Cauchy&rft.aufirst=Augustin&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Foeuvresdaugusti107caucrich%23page%2Fn27%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span> Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolland1995" class="citation cs2">Folland, Gerald B. (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c0WFd3X_R20C&q=%22Cauchy%E2%80%93Kowalevski%22"><i>Introduction to Partial Differential Equations</i></a>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-04361-2" title="Special:BookSources/0-691-04361-2"><bdi>0-691-04361-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Partial+Differential+Equations&rft.pub=Princeton+University+Press&rft.date=1995&rft.isbn=0-691-04361-2&rft.aulast=Folland&rft.aufirst=Gerald+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc0WFd3X_R20C%26q%3D%2522Cauchy%25E2%2580%2593Kowalevski%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHörmander1983" class="citation cs2">Hörmander, L. (1983), <i>The analysis of linear partial differential operators I</i>, Grundl. Math. Wissenschaft., vol. 256, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-96750-4">10.1007/978-3-642-96750-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-12104-8" title="Special:BookSources/3-540-12104-8"><bdi>3-540-12104-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0717035">0717035</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+analysis+of+linear+partial+differential+operators+I&rft.series=Grundl.+Math.+Wissenschaft.&rft.pub=Springer&rft.date=1983&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0717035%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-96750-4&rft.isbn=3-540-12104-8&rft.aulast=H%C3%B6rmander&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span> (linear case)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKashiwara1983" class="citation cs2">Kashiwara, M. (1983), <i>Systems of microdifferential equations</i>, Progress in Mathematics, vol. 34, Birkhäuser, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0817631380" title="Special:BookSources/0817631380"><bdi>0817631380</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Systems+of+microdifferential+equations&rft.series=Progress+in+Mathematics&rft.pub=Birkh%C3%A4user&rft.date=1983&rft.isbn=0817631380&rft.aulast=Kashiwara&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Kowalevsky1875" class="citation cs2">von Kowalevsky, Sophie (1875), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0080&DMDID=DMDLOG_0005">"Zur Theorie der partiellen Differentialgleichung"</a>, <i>Journal für die reine und angewandte Mathematik</i>, <b>80</b>: 1–32</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Zur+Theorie+der+partiellen+Differentialgleichung&rft.volume=80&rft.pages=1-32&rft.date=1875&rft.aulast=von+Kowalevsky&rft.aufirst=Sophie&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN243919689_0080%26DMDID%3DDMDLOG_0005&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span> (German spelling of her surname used at that time.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNakhushev2001" class="citation cs2">Nakhushev, A.M. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Cauchy–Kovalevskaya_theorem">"Cauchy–Kovalevskaya theorem"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cauchy%E2%80%93Kovalevskaya+theorem&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Nakhushev&rft.aufirst=A.M.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DCauchy%E2%80%93Kovalevskaya_theorem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACauchy%E2%80%93Kovalevskaya+theorem" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cauchy%E2%80%93Kovalevskaya_theorem&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/cauchykowalewskitheorem">PlanetMath</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Cauchy–Kovalevskaya_theorem&oldid=1184501270">https://en.wikipedia.org/w/index.php?title=Cauchy–Kovalevskaya_theorem&oldid=1184501270</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Augustin-Louis_Cauchy" title="Category:Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li><li><a href="/wiki/Category:Partial_differential_equations" title="Category:Partial differential equations">Partial differential equations</a></li><li><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Theorems in analysis</a></li><li><a href="/wiki/Category:Uniqueness_theorems" title="Category:Uniqueness theorems">Uniqueness theorems</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 November 2023, at 20:27<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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