Complex analysis is the branch of mathematics investigating holomorphic functions, i.e. functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, every holomorphic function is representable as power series in every open disc in its domain of definition, and is therefore analytic. In particular, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, such as all polynomials, the exponential function, and the trigonometric functions, are holomorphic. See also : holomorphic sheaves and vector bundles.

Subcategories

There are 5 subcategories to this category.

A

• Analytic number theory

M

• Modular forms

R

• Riemann surfaces

S

• Several complex variables
• Special functions

Articles in category "Complex analysis"

There are 90 articles in this category.

Complex analysis A Algebraic geometry and analytic geometry Analytic capacity Analytic continuation Analytic function Analytic variety Antiholomorphic function Argument principle B Blaschke product Branch point Bromwich integral C Cartan's theorems A and B Cauchy integral theorem Cauchy's integral formula Cauchy-Riemann equations Conformal map Conformally equivalent Cousin problems Critical line theorem D De Branges' theorem De Moivre's formula Dessin d'enfant Disk algebra E Edge of the wedge theorem Entire function Essential singularity Euler's formula Euler's identity Exponential function F Fundamental theorem of algebra

H H infinity Hadamard three-circle theorem Hankel contour Hardy space Hardy's theorem Harmonic conjugate Harnack's principle Holomorphic function Holomorphic sheaf Hyperfunction I Imaginary number Isolated singularity K Kramers-Kronig relations L Landau's constants Laurent series Liouville's theorem (complex analysis) List of Fourier-related transforms Logarithmic derivative M Maximum modulus principle Mellin inversion theorem Mellin transform Meromorphic function Methods of contour integration Modular form Modular function Monodromy Montel's theorem Morera's theorem N Normal family O Open mapping theorem P Path integral

P cont. Picard theorem Pole (complex analysis) Polynomial Positive-definite function Power series Principal branch Principal part Progressive function Proof that holomorphic functions are analytic R Radius of convergence Ramification Removable singularity Residue (complex analysis) Residue theorem Riemann mapping theorem Riemann surface Riemann-Roch theorem Rouché's theorem S Schwarz lemma Several complex variables Siegel upper half plane U Univalent function Upper half plane V Value distribution theory of holomorphic functions Vector bundle W Weierstrass factorization theorem Weierstrass-Casorati theorem Winding number Z Zero (complex analysis)