Fungrim home page

Fungrim entry: 499bdf

Symbol: IsHolomorphic f(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
IsHolomorphic(f(z), For(z, c)), rendered f(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c, represents the predicate that f(z)f(z) is complex differentiable in some open neighborhood of the point cc.
IsHolomorphic(f(z), ForElement(z, S)), rendered f(z) is holomorphic on zSf(z) \text{ is holomorphic on } z \in S, represents the predicate that f(z)f(z) is complex differentiable in some open neighborhood of every point in the set SS.
As a special case f(z) is holomorphic at z=~f(z) \text{ is holomorphic at } z = {\tilde \infty} is equivalent to f ⁣(1z) is holomorphic at z=0f\!\left(\frac{1}{z}\right) \text{ is holomorphic at } z = 0.
As a special case f(z) is holomorphic at z=if(z) \text{ is holomorphic at } z = i \infty represents the predicate that f(z)f(z) is a periodic function on the upper half plane that is holomorphic at infinity (in the sense of modular function theory)
Definitions:
Fungrim symbol Notation Short description
IsHolomorphicf(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
UnsignedInfinity~{\tilde \infty} Unsigned infinity
ConstIii Imaginary unit
Infinity\infty Positive infinity
Source code for this entry:
Entry(ID("499bdf"),
    SymbolDefinition(IsHolomorphic, IsHolomorphic(f(z), For(z, c)), "Holomorphic predicate"),
    Description(SourceForm(IsHolomorphic(f(z), For(z, c))), ", rendered", IsHolomorphic(f(z), For(z, c)), ", represents the predicate", "that", f(z), "is complex differentiable in some open neighborhood of the point", c, "."),
    Description(SourceForm(IsHolomorphic(f(z), ForElement(z, S))), ", rendered", IsHolomorphic(f(z), ForElement(z, S)), ", represents the predicate", "that", f(z), "is complex differentiable in some open neighborhood of every point in the set", S, "."),
    Description("As a special case", IsHolomorphic(f(z), For(z, UnsignedInfinity)), " is equivalent to", IsHolomorphic(f(Div(1, z)), For(z, 0)), "."),
    Description("As a special case", IsHolomorphic(f(z), For(z, Mul(ConstI, Infinity))), " represents the predicate that", f(z), "is a periodic function on the upper half plane that is holomorphic at infinity (in the sense of modular function theory)"))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC