Fungrim entry: 231a99

Symbol: ComplexZeroMultiplicity —

\mathop{\operatorname{ord}}\limits_{z=c} f(z)

— Multiplicity (order) of complex zero

ComplexZeroMultiplicity(f(z), For(z, c)), rendered

\mathop{\operatorname{ord}}\limits_{z=c} f(z)

, gives the root multiplicity (order of vanishing) of

f(z)

at the point

z = c

f

is holomorphic at

c

and

f(c) \ne 0

, the multiplicity is zero.

z = c

is a pole of

f(z)

, returns

-n

where

n

is the order of the pole.

In other words, this operator returns the order of the first nonzero term in the Laurent series of

f(z)

z = c

In the special case where

f(z) = 0

in a neighborhood of

c

, the order is

\infty

The result is undefined if

f(z)

is not meromorphic at

c

The special expression For(z, c) declares z as a locally bound variable within the scope of the arguments to this operator.

Definitions:

Fungrim symbol	Notation	Short description
`ComplexZeroMultiplicity`	$\mathop{\operatorname{ord}}\limits_{z=c} f(z)$	Multiplicity (order) of complex zero
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("231a99"),
    SymbolDefinition(ComplexZeroMultiplicity, ComplexZeroMultiplicity(f(z), For(z, c)), "Multiplicity (order) of complex zero"),
    Description(SourceForm(ComplexZeroMultiplicity(f(z), For(z, c))), ", rendered", ComplexZeroMultiplicity(f(z), For(z, c)), ", gives the root multiplicity (order of vanishing) of", f(z), "at the point", Equal(z, c), "."),
    Description("If", f, "is holomorphic at", c, "and", NotEqual(f(c), 0), ", the multiplicity is zero."),
    Description("If", Equal(z, c), "is a pole of", f(z), ", returns", Neg(n), "where", n, "is the order of the pole."),
    Description("In other words, this operator returns the order of the first nonzero term in the Laurent series of", f(z), "at", Equal(z, c), "."),
    Description("In the special case where", Equal(f(z), 0), "in a neighborhood of", c, ", the order is", Infinity, "."),
    Description("The result is undefined if", f(z), "is not meromorphic at", c, "."),
    Description("The special expression", SourceForm(For(z, c)), "declares", SourceForm(z), "as a locally bound variable within the scope of the arguments to this operator."))

Topics using this entry

Operators