# Fungrim entry: b4b319

Symbol: ComplexDerivative $\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
The call syntax for this operator is the same as for Derivative.
The result is defined as $f'(z) = \lim_{h \to 0} \frac{f\!\left(z + h\right) - f(z)}{h}$ where the limit is taken with respect to a complex variable $h$ ( ComplexLimit ).
If this limit exists (and is finite), then $f$ is holomorphic at $z$.
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Derivative$\frac{d}{d z}\, f\!\left(z\right)$ Derivative
ComplexLimit$\lim_{z \to a} f(z)$ Limiting value, complex variable
Source code for this entry:
Entry(ID("b4b319"),
SymbolDefinition(ComplexDerivative, ComplexDerivative(Call(f, z), For(z, z)), "Complex derivative"),
Description("The call syntax for this operator is the same as for", SourceForm(Derivative), "."),
Description("The result is defined as", Equal(ComplexDerivative(f(z), For(z, z)), ComplexLimit(Div(Sub(f(Add(z, h)), f(z)), h), For(h, 0))), "where the limit is taken with respect to a complex variable", h, "(", SourceForm(ComplexLimit), ")."),
Description("If this limit exists (and is finite), then", f, "is holomorphic at", z, "."))

## Topics using this entry

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