`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:

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