Fungrim entry: 0895b1

Symbol: IsMeromorphic $f(z) \text{ is meromorphic at } z = c$ Meromorphic predicate
IsMeromorphic(f(z), For(z, c)), rendered $f(z) \text{ is meromorphic at } z = c$, represents the predicate that $f(z)$ is meromorphic in some open neighborhood of the point $c$.
IsMeromorphic(f(z), ForElement(z, S)), rendered $f(z) \text{ is meromorphic on } z \in S$, represents the predicate that $f(z)$ is meromorphic in some open neighborhood of every point in the set $S$.
As a special case $f(z) \text{ is meromorphic at } z = {\tilde \infty}$ is equivalent to $f\!\left(\frac{1}{z}\right) \text{ is meromorphic at } z = 0$.
As a special case $f(z) \text{ is meromorphic at } z = i \infty$ represents the predicate that $f(z)$ is a periodic function on the upper half plane that is meromorphic at infinity (in the sense of modular function theory)
Definitions:
Fungrim symbol Notation Short description
IsMeromorphic$f(z) \text{ is meromorphic at } z = c$ Meromorphic predicate
UnsignedInfinity${\tilde \infty}$ Unsigned infinity
ConstI$i$ Imaginary unit
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("0895b1"),
SymbolDefinition(IsMeromorphic, IsMeromorphic(f(z), For(z, c)), "Meromorphic predicate"),
Description(SourceForm(IsMeromorphic(f(z), For(z, c))), ", rendered", IsMeromorphic(f(z), For(z, c)), ", represents the predicate", "that", f(z), "is meromorphic in some open neighborhood of the point", c, "."),
Description(SourceForm(IsMeromorphic(f(z), ForElement(z, S))), ", rendered", IsMeromorphic(f(z), ForElement(z, S)), ", represents the predicate", "that", f(z), "is meromorphic in some open neighborhood of every point in the set", S, "."),
Description("As a special case", IsMeromorphic(f(z), For(z, UnsignedInfinity)), " is equivalent to", IsMeromorphic(f(Div(1, z)), For(z, 0)), "."),
Description("As a special case", IsMeromorphic(f(z), For(z, Mul(ConstI, Infinity))), " represents the predicate that", f(z), "is a periodic function on the upper half plane that is meromorphic at infinity (in the sense of modular function theory)"))

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

2021-03-15 19:12:00.328586 UTC