FunctionMeromorphic

True if f is meromorphic (a ratio of entire functions, poles allowed) everywhere, False otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFunctionMeromorphic✓↗

FunctionMeromorphic(f, x)whether f, viewed as a function of a complex variable, is meromorphic on the whole plane.

Typed signatures

Type	Package	Applies
(expression, symbol) -> expression	analytic	always

Domain: Elementary functions

Details

• A polynomial, Exp/Sin/Cos of an affine argument (entire, hence trivially meromorphic), a genuine rational function, and Tan ($=\sin/\cos$, a ratio of entire functions) are all meromorphic; Sqrt and Ln are not, since a branch point is not a pole.
• Beyond those shapes the head reads sums, products, quotients and integer powers of meromorphic functions, with Exp, Sin, Cos, Sinh and Cosh of an entire argument, and Tan, Sec, Sech and the other reciprocals of one: x/(x^2+1) and Sec(x) are True. It says nothing about an essential singularity such as Exp(1/x).
• This is a strictly larger class than FunctionAnalytic — the one distinction the two heads draw is exactly the poles a rational function or Tan has, which disqualify it from FunctionAnalytic but not from this one.

Examples

Basic #

A rational function is meromorphic even though it isn't entire

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In$\mathrm{FunctionMeromorphic}(\frac{1}{x}, x)$Out$\mathrm{True}$

Known value · Wolfram 15.0.0, hand cross-checked

Tangent is sine over cosine, a ratio of entire functions

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In$\mathrm{FunctionMeromorphic}(\tan(x), x)$Out$\mathrm{True}$

Known value · Wolfram 15.0.0, hand cross-checked

A branch point disqualifies log even though it has no poles

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In$\mathrm{FunctionMeromorphic}(\ln(x), x)$Out$\mathrm{False}$

Known value · Wolfram 15.0.0, hand cross-checked

Polynomials are entire, hence meromorphic

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In$\mathrm{FunctionMeromorphic}(x^2, x)$Out$\mathrm{True}$

Scope #

The square root has a branch point, so it is not meromorphic

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In$\mathrm{FunctionMeromorphic}(\sqrt{z}, z)$Out$\mathrm{False}$

Applications #

The logarithm has a branch cut, so is not meromorphic

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In$\mathrm{FunctionMeromorphic}(\ln(z), z)$Out$\mathrm{False}$

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In$\bigl\lbrack\mathrm{FunctionMeromorphic}(\frac{x}{x^2+1}, x), \mathrm{FunctionMeromorphic}(\frac{x^2+1}{x^2-1}, x), \mathrm{FunctionMeromorphic}(x^3-2x^2-5x+6, x)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack$

live ↗

In$\bigl\lbrack\mathrm{FunctionMeromorphic}(\tan(x), x), \mathrm{FunctionMeromorphic}(\sec(x), x), \mathrm{FunctionMeromorphic}(\mathrm{sech}(x), x)\bigr\rbrack$Out$\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack$

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In$\mathrm{FunctionMeromorphic}(\tan(z)+\sin(z), z)$Out$\mathrm{True}$

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In$\mathrm{FunctionMeromorphic}(\cot(z)-\frac{z^2}{z+1}, z)$Out$\mathrm{True}$

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In$\mathrm{FunctionMeromorphic}(\frac{z^3\tanh(z)}{z^2+1}, z)$Out$\mathrm{True}$

About

Examples	11
Checked against	Wolfram Language 11 agree
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