# Fungrim entry: 499bdf

Symbol: IsHolomorphic $f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
IsHolomorphic(f(z), For(z, c)), rendered $f(z) \text{ is holomorphic at } z = c$, represents the predicate that $f(z)$ is complex differentiable in some open neighborhood of the point $c$.
IsHolomorphic(f(z), ForElement(z, S)), rendered $f(z) \text{ is holomorphic on } z \in S$, represents the predicate that $f(z)$ is complex differentiable in some open neighborhood of every point in the set $S$.
As a special case $f(z) \text{ is holomorphic at } z = {\tilde \infty}$ is equivalent to $f\!\left(\frac{1}{z}\right) \text{ is holomorphic at } z = 0$.
As a special case $f(z) \text{ is holomorphic at } z = i \infty$ 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)
Definitions:
Fungrim symbol Notation Short description
IsHolomorphic$f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
UnsignedInfinity${\tilde \infty}$ Unsigned infinity
ConstI$i$ 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

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2020-04-08 16:14:44.404316 UTC