Given a Z #manifold M, a differential of first kind ω is same thing as a 1-form that is everywhere #holomorphic; on an algebraic variety V that is non-singular it would be a global section of coherent #sheaf Ω1 of #Kähler differentials. Via abelian integrals.

- classical Paley–Wiener theorems make use of the #holomorphic Fourier transform on classes of square-integrable functions supported on the real line.
Rudin

- inverse of a #holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a global analytic function.

AdS/CFT <->> duality bw pure #quantum gravity g in (2+1)-d anti #deSitter space and extremal #holomorphic CFTs. g has no local degrees of freedom, but when the cosmological constant is negative, there is nontrivial content in theory, due to existence of BTZ #blackhole solutions.

#deRham's thesis, Hodge realized that real and imaginary parts of a #holomorphic 1-form on a #Riemann surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now #Hodge star operator

#Hodge on #Lefschetz's paper: similar principles applied to algebraic surfaces:
if ω =/=0 #holomorphic form on an algebraic surface, then i: sqrt {-1}}\,\omega ^ {\bar {\omega } is + , i^2=/=0, must b. =>ω => =/=0 #cohomology class,so its periods=/=0
Fixed a question of #Severi

#deRham's thesis, Hodge realized that real and imaginary parts of a #holomorphic 1-form on a #Riemann surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now #Hodge star operator

#Hodge on #Lefschetz's paper: similar principles applied to algebraic surfaces:
if ω =/=0 #holomorphic form on an algebraic surface, then i: sqrt {-1}}\,\omega ^ {\bar {\omega } is + , i^2=/=0, must b. =>ω => =/=0 #cohomology class,so its periods=/=0
Fixed a question of #Severi

A #holomorphic function need not ,possess an antiderivative on its domain, w/o additional assumptions. converse does hold e.g. if domain is simply connected; this is #Cauchy's integral theorem, stating that line integral of a holomorphic function along a closed curve is zero.

A #holomorphic function need not ,possess an antiderivative on its domain, w/o additional assumptions. converse does hold e.g. if domain is simply connected; this is #Cauchy's integral theorem, stating that line integral of a holomorphic function along a closed curve is zero.

If δ is not given, it is assumed to be π, and the sector is in fact the whole plane ℂ, with the exception of a half-line originating at z = 0 and pointing into the direction of −α, usually serving as a branch cut.
a single-valued and #holomorphic logarithm

If δ is not given, it is assumed to be π, and the sector is in fact the whole plane ℂ, with the exception of a half-line originating at z = 0 and pointing into the direction of −α, usually serving as a branch cut.
a single-valued and #holomorphic logarithm

By open mapping theorem : non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of #holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

By open mapping theorem : non-constant holomorphic function cannot map an open disk onto a portion of any line embedded in the complex plane. Images of #holomorphic functions can be of real dimension zero (if constant) or two (if non-constant) but never of dimension 1.

for #holomorphic dynamical systems, the orbits are #Riemann surfaces.

for #holomorphic dynamical systems, the orbits are #Riemann surfaces.

hyperfunction #holomorphic functions used as test functions refined theory: #MikioSato's algebraic analysis, using #sheaftheory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example #Feynman integrals.

hyperfunction #holomorphic functions used as test functions refined theory: #MikioSato's algebraic analysis, using #sheaftheory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example #Feynman integrals.