Fungrim symbol: ComplexDerivative

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\!\left(z\right)}{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`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable

Source code for this entry:

Entry(ID("b4b319"),
    SymbolDefinition(ComplexDerivative, ComplexDerivative(Call(f, z), 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), z, z), ComplexLimit(Div(Sub(f(Add(z, h)), f(z)), h), 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 symbol

The symbol ComplexDerivative appears in 1 topics:

Operators (in 4 entries)

Entries using this symbol

The symbol ComplexDerivative appears in 4 entries: