FunctionAnalytic

True if f is analytic (entire) at every real x, False otherwise.

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FunctionAnalytic(f, x)whether f, viewed as a function of a complex variable, is entire.

Typed signatures

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

Domain: Elementary functions

Details

Confirmed against wolframscript, this is stricter than "analytic on its real domain" -- FunctionAnalytic(1/x, x) is False even though $1/x$ is perfectly analytic away from 0, because it is not analytic EVERYWHERE (it has a pole at $x=0$ , real domain or not).
True for a polynomial and Exp/Sin/Cos of an affine argument (all entire); False for any genuine rational function (a nonconstant denominator always has a complex pole somewhere), Sqrt and Ln (a branch point), and Tan (poles where cosine vanishes).

Examples

Basic #

Polynomials are entire

\mathrm{FunctionAnalytic}(x^2, x)

Out

\mathrm{True}

A pole anywhere in the complex plane disqualifies it, real domain or not

\mathrm{FunctionAnalytic}(\frac{1}{x}, x)

Out

\mathrm{False}

Known value · Wolfram 15.0.0, hand cross-checked

Sqrt has a branch point at the origin

\mathrm{FunctionAnalytic}(\sqrt{x}, x)

Out

\mathrm{False}

Known value · Wolfram 15.0.0, hand cross-checked

Sine is entire

\mathrm{FunctionAnalytic}(\sin(x), x)

Out

\mathrm{True}

Known value · Wolfram 15.0.0, hand cross-checked

Tangent has poles, so it is not analytic everywhere

\mathrm{FunctionAnalytic}(\tan(x), x)

Out

\mathrm{False}

Applications #

Polynomials are analytic

\bigl\lbrack\mathrm{FunctionAnalytic}(x, x), \mathrm{FunctionAnalytic}(x^2+1, x), \mathrm{FunctionAnalytic}(x^3-2x^2-5x+6, x)\bigr\rbrack

Out

\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack

Sine, cosine and exp are entire, so analytic

\bigl\lbrack\mathrm{FunctionAnalytic}(\sin(x), x), \mathrm{FunctionAnalytic}(\cos(x), x), \mathrm{FunctionAnalytic}(e^{x}, x)\bigr\rbrack

Out

\bigl\lbrack\mathrm{True}, \mathrm{True}, \mathrm{True}\bigr\rbrack

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