Functions of a complex variable Basic notions Concept
Holomorphic function
Summary
A holomorphic function is a complex function that is complex differentiable at each point of an open subset of , see the definition. Holomorphic functions are the main objects of complex analysis. They are infinitely differentiable and they are equal to their Taylor series near each point.
Where this leads
79 pages build on this one.
- Conformal invariance of harmonic functions
- Malgrange preparation theorem
- Riemann zeta function as a series
- Matrix Wiener–Hopf factorization
- Bloch space
- External ray
- Weierstrass preparation theorem
- Beltrami equation
- Hyperfunction
- Matrix Riemann–Hilbert problem
- Stein interpolation theorem
- Log-gamma function
- Schwarz integral formula
- Gamma function as Mellin transform of the exponential
- Theta function
- Hilbert's twenty-first problem
- Log-modulus of a holomorphic function is subharmonic
- Cauchy-type integral
- Holomorphy on Banach spaces
- Bernstein's theorem on analytic functions
- Holomorphic functional calculus
- Herglotz representation theorem
- Poisson–Mellin–Newton cycle
- Wiener–Hopf factorization
- Poincaré's functional equation
- Local existence of harmonic conjugates
- Ahlfors–Bers theorem
- Differentiability of the Mellin transform
- Modality of a singularity
- Hartogs's theorem on separate holomorphicity
- Stieltjes transform
- Hypergeometric function
- Cauchy–Kovalevskaya theorem for ODEs
- Dirichlet space
- Minimal harmonic function
- Complex dynamics
- Koenigs linearization theorem
- Analytic semigroup
- Singularity theory
- Bergman space
- Stokes phenomenon
- Abel–Plana formula
- Bergman kernel
- Dedekind eta function
- Arnold's classification of simple critical points
- Modular form
- Wiener–Hopf method
- Dolbeault operator
- Littlewood subordination theorem
- Cauchy–Riemann operator
- Tjurina number
- Analyticity of the bilateral Laplace transform of a law
- Harmonic functions on Riemann surfaces
- Riemann–Hilbert problem
- Herglotz–Riesz kernel
- Koenigs function
- Monodromy
- Carleman's singular integral equation
- Liouville's theorem for planar harmonic functions
- Mather's stability theorem
- Topological modular forms
- Resolvent
- Hadamard three-circle theorem for subharmonic functions
- Bürmann–Lagrange series
- Riemann zeta function
- Measurable Riemann mapping theorem
- Pseudoanalytic function
- Wielandt's theorem
- Bateman transform
- Method of steepest descent
- Parabolic and hyperbolic Riemann surfaces
- Hadamard's two-constants theorem
- Hardy space
- Böttcher's equation
- Nevanlinna parametrization of solutions
- Weierstrass–Enneper parameterization
- Euler integral for the gamma function
- Holomorphy of the gamma function
- Ramanujan's master theorem
Further reading
- Web https://en.wikipedia.org/wiki/Holomorphic_function, lead section
About this page
| Tag | 0NAHXR1FRH |
|---|---|
| URI | m2://0NAHXR1FRH |
| Aliases | holomorphic, complex-analytic-function |
| Kind | concept |
| Writing | W1 stated |
| MSC | 30A99 |
| Specializes | Meromorphic function |
| Same as |
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What links here
Definitions
Special cases
Used by
- Conformal invariance of harmonic functions
- Bloch space
- Weierstrass preparation theorem
- Beltrami equation
- Hyperfunction
- Matrix Riemann–Hilbert problem
- Stein interpolation theorem
- Log-gamma function
- Schwarz integral formula
- Theta function
- Log-modulus of a holomorphic function is subharmonic
- Cauchy-type integral
- Holomorphy on Banach spaces
- Bernstein's theorem on analytic functions
- Holomorphic functional calculus
- Herglotz representation theorem
- Poisson–Mellin–Newton cycle
- Wiener–Hopf factorization
- Poincaré's functional equation
- Local existence of harmonic conjugates
- Ahlfors–Bers theorem
- Differentiability of the Mellin transform
- Modality of a singularity
- Hartogs's theorem on separate holomorphicity
- Stieltjes transform
- Hypergeometric function
- Cauchy–Kovalevskaya theorem for ODEs
- Dirichlet space
- Complex dynamics
- Koenigs linearization theorem
- Analytic semigroup
- Singularity theory
- Bergman space
- Stokes phenomenon
- Abel–Plana formula
- Dedekind eta function
- Arnold's classification of simple critical points
- Modular form
- Wiener–Hopf method
- Dolbeault operator
- Littlewood subordination theorem
- Cauchy–Riemann operator
- Tjurina number
- Analyticity of the bilateral Laplace transform of a law
- Harmonic functions on Riemann surfaces
- Riemann–Hilbert problem
- Herglotz–Riesz kernel
- Koenigs function
- Monodromy
- Resolvent
- Bürmann–Lagrange series
- Riemann zeta function
- Pseudoanalytic function
- Wielandt's theorem
- Bateman transform
- Method of steepest descent
- Hadamard's two-constants theorem
- Hardy space
- Böttcher's equation
- Nevanlinna parametrization of solutions
- Weierstrass–Enneper parameterization
- Euler integral for the gamma function
- Holomorphy of the gamma function
- Ramanujan's master theorem