FunctionSingularities

Where f fails to be analytic, as a condition in x.

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FunctionSingularities(f, x)the set of real x where f is not analytic (a pole, or outside its domain, including its branch points).

Typed signatures

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

Domain: Elementary functions

Details

• Not simply the complement of FunctionDomain -- confirmed against wolframscript, Sqrt and Log are singular AT their boundary point too, even though the real value there is defined ($\sqrt0=0$): FunctionSingularities(Sqrt(x), x) is x<=0, not the strict x<0 the domain's complement would give.
• A polynomial or Exp has none (False); a rational function's are the real zeros of its denominator; Sqrt/Ln's are the closed complement of where their argument is positive; Tan's are Cos(x)==0.

Examples

Basic #

A simple pole at the origin

live ↗

In$\mathrm{FunctionSingularities}(\frac{1}{x}, x)$Out$x=0$

Known value · Wolfram 15.0.0, hand cross-checked

Sqrt is singular AT x=0 too, not just where the radicand goes negative

live ↗

In$\mathrm{FunctionSingularities}(\sqrt{x}, x)$Out$x≤0$

Known value · Wolfram 15.0.0, hand cross-checked

A polynomial is entire

live ↗

In$\mathrm{FunctionSingularities}(x^2, x)$Out$\mathrm{False}$

Known value · Wolfram 15.0.0, hand cross-checked

Log's branch cut and its excluded domain coincide here

live ↗

In$\mathrm{FunctionSingularities}(\ln(x), x)$Out$x≤0$

Tangent is singular where cosine vanishes

live ↗

In$\mathrm{FunctionSingularities}(\tan(x), x)$Out$\cos(x)=0$

About

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