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Fungrim entry: 231a99

Symbol: ComplexZeroMultiplicity ordz=cf(z)\mathop{\operatorname{ord}}\limits_{z=c} f(z) Multiplicity (order) of complex zero
ComplexZeroMultiplicity(f(z), For(z, c)), rendered ordz=cf(z)\mathop{\operatorname{ord}}\limits_{z=c} f(z), gives the root multiplicity (order of vanishing) of f(z)f(z) at the point z=cz = c.
If ff is holomorphic at cc and f(c)0f(c) \ne 0, the multiplicity is zero.
If z=cz = c is a pole of f(z)f(z), returns n-n where nn 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)f(z) at z=cz = c.
In the special case where f(z)=0f(z) = 0 in a neighborhood of cc, the order is \infty.
The result is undefined if f(z)f(z) is not meromorphic at cc.
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
ComplexZeroMultiplicityordz=cf(z)\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."))

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2020-01-31 18:09:28.494564 UTC