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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikiquote</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b><a href="https://en.wikipedia.org/wiki/Complex_analysis" class="extiw" title="w:Complex analysis">Complex analysis</a></b>, traditionally known as the <b>theory of functions of a complex variable</b>, is the branch of mathematical analysis that investigates functions of <a href="https://en.wikipedia.org/wiki/complex_numbers" class="extiw" title="w:complex numbers">complex numbers</a>. </p> <table role="presentation" class="metadata plainlinks stub" style="background: transparent; font-style: italic;"><tbody><tr> <td class="noresize"><span typeof="mw:File"><a href="/wiki/File:E-to-the-i-pi.svg" class="mw-file-description"><img alt="Stub icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/34px-E-to-the-i-pi.svg.png" decoding="async" width="34" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/51px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/68px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span></td> <td>This <a href="/wiki/Category:Mathematics" title="Category:Mathematics">mathematics</a>-related article is a <a href="/wiki/Wikiquote:Stub" title="Wikiquote:Stub">stub</a>. You can help out with Wikiquote by <a class="external text" href="https://en.wikiquote.org/w/index.php?title=Complex_analysis&action=edit">expanding</a> it!</td> </tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Quotes">Quotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Complex_analysis&action=edit&section=1" title="Edit section: Quotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Le d茅veloppement naturel de cette 茅tude conduisit bient么t les g茅om猫tres 脿 embrasser dans leurs recherches les valeurs imaginaires de la variable aussi bien que les valeurs r茅elles. La th茅orie de la s茅rie de Taylor, celle des fonctions elliptiques, la vaste doctrine de Cauchy firent 茅clater la f茅condit茅 de cette g茅n茅ralisation. Il apparut que, <b>entre deux v茅rit茅s du domaine r茅el, le chemin le plus facile et le plus court passe bien souvent par le domaine complexe.</b></i> <ul><li>The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, <b>between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain.</b> <ul><li><cite style="font-style:normal" class="book" id="CITEREFPaul_Painlev茅1967">Paul Painlev茅 (1967) [1900]. <i>Analyse des travaux scientifiques</i>. Gauthier-Villars (reprinted by Librairie Scientifique et Technique, Albert Blanchard). <a href="/wiki/Special:BookSources/2853671313" title="Special:BookSources/2853671313">ISBN 2853671313</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analyse+des+travaux+scientifiques&rft.aulast=Paul+Painlev%C3%A9&rft.au=Paul+Painlev%C3%A9&rft.date=1967&rft.pub=Gauthier-Villars+%28reprinted+by+Librairie+Scientifique+et+Technique%2C+Albert+Blanchard%29&rft.isbn=2853671313&rfr_id=info:sid/en.wikipedia.org:Complex_analysis"><span style="display: none;"> </span></span></li></ul></li></ul></li></ul> <ul><li>At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> holds. <ul><li><cite style="font-style:normal" class="book" id="CITEREFTheodore_Gamelin2003">Theodore Gamelin (17 July 2003). <i><a rel="nofollow" class="external text" href="http://books.google.com/books?id=61BO2UY9QN4C&pg=PA370">Complex Analysis</a></i>. Springer Science & Business Media. pp. 370. <a href="/wiki/Special:BookSources/978-0-387-95069-3" title="Special:BookSources/978-0-387-95069-3">ISBN 978-0-387-95069-3</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+Analysis&rft.aulast=Theodore+Gamelin&rft.au=Theodore+Gamelin&rft.date=17+July+2003&rft.pages=pp.%26nbsp%3B370&rft.pub=Springer+Science+%26+Business+Media&rft.isbn=978-0-387-95069-3&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D61BO2UY9QN4C%26pg%3DPA370&rfr_id=info:sid/en.wikipedia.org:Complex_analysis"><span style="display: none;"> </span></span></li></ul></li></ul> <ul><li>The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, <a href="/wiki/Number_theory" title="Number theory">number theory</a>, which is the study of the discrete, and complex analysis, which deals with continuous processes. <ul><li><cite style="font-style:normal" class="book" id="CITEREFDavid_M._Burton2006">David M. Burton (1 May 2006). <i><a rel="nofollow" class="external text" href="http://books.google.com/books?id=XMQjuoTqqRMC">Elementary Number Theory</a></i>. Tata McGraw-Hill Publishing Company Limited. pp. 376. <a href="/wiki/Special:BookSources/978-0-07-061607-3" title="Special:BookSources/978-0-07-061607-3">ISBN 978-0-07-061607-3</a>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory&rft.aulast=David+M.+Burton&rft.au=David+M.+Burton&rft.date=1+May+2006&rft.pages=pp.%26nbsp%3B376&rft.pub=Tata+McGraw-Hill+Publishing+Company+Limited&rft.isbn=978-0-07-061607-3&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXMQjuoTqqRMC&rfr_id=info:sid/en.wikipedia.org:Complex_analysis"><span style="display: none;"> </span></span></li></ul></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=Complex_analysis&action=edit&section=2" title="Edit section: External links"><span>edit</span></a><span 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<a href="/wiki/Kurt_G%C3%B6del" title="Kurt G枚del">G枚del</a> • <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Grassmann</a> • <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck</a> • <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a> • <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a> • <a href="/wiki/Hypatia" title="Hypatia">Hypatia</a> • <a href="/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Lagrange</a> • <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a> • <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Leibniz</a> • <a href="/wiki/John_Milnor" title="John Milnor">Milnor</a> • <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> • <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> • <a href="/wiki/Emmy_Noether" title="Emmy Noether">Noether</a> • <a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a> • <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Perelman</a> • <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincar茅">Poincar茅</a> • <a href="/wiki/George_P%C3%B3lya" title="George P贸lya">P贸lya</a> • <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> • <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a> • <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> • <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Schwartz</a> • <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre</a> • <a href="/wiki/Terence_Tao" title="Terence Tao">Tao</a> • <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski</a> • <a href="/wiki/Thales" title="Thales">Thales</a> • <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a> • <a href="/wiki/Andr%C3%A9_Weil" title="Andr茅 Weil">Weil</a> • <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a> • <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Wiles</a> • <a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a> </p> </td></tr> <tr> <td><a href="/wiki/Number" title="Number">Numbers</a> </td> <td style="font-size:0.8em"> <p><a href="/wiki/1_(number)" title="1 (number)">1</a> • <a href="/wiki/23" title="23">23</a> • <a href="/wiki/360_(number)" title="360 (number)">360</a> • <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><i>e</i></a> • <a href="/wiki/%CE%A0" title="螤"><i>蟺</i></a> • <a href="/wiki/Fibonacci_numbers" title="Fibonacci numbers">Fibonacci numbers</a> • <a href="/wiki/Irrational_number" title="Irrational number">Irrational number</a> • <a href="/wiki/Negative_number" title="Negative number">Negative number</a> • <a href="/wiki/Number" title="Number">Number</a> • <a href="/wiki/Prime_number" title="Prime number">Prime number</a> • <a href="/wiki/Quaternion" title="Quaternion">Quaternion</a> • <a href="/wiki/Octonion" title="Octonion">Octonion</a> </p> </td></tr> <tr> <td>Concepts </td> <td style="font-size:0.8em"> <p><a href="/wiki/Abstraction_(mathematics)" title="Abstraction (mathematics)">Abstraction</a> • <a href="/wiki/Algorithms" title="Algorithms">Algorithms</a> • <a href="/wiki/Axiomatic_system" title="Axiomatic system">Axiomatic system</a> • <a href="/wiki/Completeness" title="Completeness">Completeness</a> • <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deductive reasoning</a> • <a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> • <a href="/wiki/Dimension" title="Dimension">Dimension</a> • <a href="/wiki/Ellipse" title="Ellipse">Ellipse</a> • <a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a> • <a href="/wiki/Exponential_growth" title="Exponential growth">Exponential growth</a> • <a href="/wiki/Infinity" title="Infinity">Infinity</a> • <a href="/wiki/Integration" title="Integration">Integration</a> • <a href="/wiki/Geodesic" title="Geodesic">Geodesic</a> • <a href="/wiki/Mathematical_induction" title="Mathematical induction">Induction</a> • <a href="/wiki/Mathematical_proof" title="Mathematical proof">Proof</a> • <a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a> • <a href="/wiki/Principle_of_least_action" title="Principle of least action">Principle of least action</a> • <a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a> • <a href="/wiki/Probability" title="Probability">Probability</a> • <a href="/wiki/Randomness" title="Randomness">Randomness</a> • <a href="/wiki/Theorem" title="Theorem">Theorem</a> • <a href="/wiki/Topological_space" title="Topological space">Topological space</a> • <a href="/wiki/Wave_equation" title="Wave equation">Wave equation</a> </p> </td></tr> <tr> <td>Results </td> <td style="font-size:0.8em"> <p><a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a> • <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> </p> </td></tr> <tr> <td><a href="/wiki/Pure_mathematics" title="Pure mathematics">Pure math</a> </td> <td style="font-size:0.8em"> <p><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a> • <a href="/wiki/Algebra" title="Algebra">Algebra</a> • <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a> • <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> (<a href="/wiki/Sheaf_theory" title="Sheaf theory">Sheaf theory</a>) • <a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic topology</a> • <a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a> • <a href="/wiki/Calculus" title="Calculus">Calculus</a> • <a href="/wiki/Category_theory" title="Category theory">Category theory</a> • <a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> • <a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a> • <a class="mw-selflink selflink">Complex analysis</a> • <a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a> • <a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a> • <a href="/wiki/Differential_topology" title="Differential topology">Differential topology</a> • <a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a> • <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a> • <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> • <a href="/wiki/Game_theory" title="Game theory">Game theory</a> • <a href="/wiki/Geometry" title="Geometry">Geometry</a> • <a href="/wiki/Global_analysis" title="Global analysis">Global analysis</a> • <a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> • <a href="/wiki/Group_theory" title="Group theory">Group theory</a> • <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a> • <a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a> • <a href="/wiki/Invariant_theory" title="Invariant theory">Invariant theory</a> • <a href="/wiki/Mathematical_logic" title="Mathematical logic">Logic</a> • <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean geometry</a> • <a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a> • <a href="/wiki/Number_theory" title="Number theory">Number theory</a> • <a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a> • <a href="/wiki/Operations_research" title="Operations research">Operations research</a> • <a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a> • <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a> • <a href="/wiki/Set_theory" title="Set theory">Set theory</a> • <a href="/wiki/Sheaf_theory" title="Sheaf theory">Sheaf theory</a> • <a href="/wiki/Statistics" title="Statistics">Statistics</a> • <a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic geometry</a> • <a href="/wiki/Topology" title="Topology">Topology</a> </p> </td></tr> <tr> <td><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied math</a> </td> <td style="font-size:0.8em"> <p><a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">Computational fluid dynamics</a> • <a href="/wiki/Econometrics" title="Econometrics">Econometrics</a> • <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a> • <a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a> • <a href="/wiki/Science" title="Science">Science</a> </p> </td></tr> <tr> <td><a href="/wiki/History_of_mathematics" title="History of mathematics">History of math</a> </td> <td style="font-size:0.8em"> <p><a href="/wiki/Ancient_Greek_mathematics" title="Ancient Greek mathematics">Ancient Greek mathematics</a> • <a href="/wiki/Euclid%E2%80%99s_Elements" title="Euclid鈥檚 Elements">Euclid's <i>Elements</i></a> • <a href="/wiki/History_of_algebra" title="History of algebra">History of algebra</a> • <a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a> • <a href="/wiki/History_of_logarithms" title="History of logarithms">History of logarithms</a> • <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian mathematics</a> • <a href="/wiki/Principia_Mathematica" title="Principia Mathematica"><i>Principia Mathematica</i></a> </p> </td></tr> <tr> <td>Other </td> <td style="font-size:0.8em"> <p><a href="/wiki/Mathematics_and_mysticism" title="Mathematics and mysticism">Mathematics and mysticism</a> • <a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a> • <a href="/wiki/Mathematics,_from_the_points_of_view_of_the_Mathematician_and_of_the_Physicist" title="Mathematics, from the points of view of the Mathematician and of the Physicist">Mathematics, from the points of view of the Mathematician and of the Physicist</a> • <a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a> • <a href="/wiki/Unification_in_science_and_mathematics" title="Unification in science and mathematics">Unification in science and mathematics</a> </p> </td></tr></tbody></table> <p><br clear="all" /> </p>    </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" 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