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 $z = c$.

If $f$
is holomorphic at $c$
and $f(c) \ne 0$, the multiplicity is zero.

If $z = c$
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 $f(z)$
at $z = c$.

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."))