ExpToTrig

Rewrite every exponential in expr as circular or hyperbolic functions -- the inverse of TrigToExp. Provided by @enumeratio/analytic.

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ExpToTrig(expr)expr with each Exp rewritten via Euler's formula or its hyperbolic analogue.

Typed signatures

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

Domain: Transformations

Details

Rewrites Exp(ix) to cos(x) + i·sin(x) and a real Exp(x) to cosh(x) + sinh(x) everywhere in the tree, then simplifies. A combination that cancels back down to a single Cosh/Sinh/Sin -- the Scope examples -- falls out of that simplification pass; there is no separate 'recognize this shape' rule.

Examples

Basic #

Euler's formula

In

\mathrm{ExpToTrig}(\exp(i x))

Out

i\sin(x)+\cos(x)

A real exponential splits into hyperbolic parts

In

\mathrm{ExpToTrig}(e^{x})

Out

\sinh(x)+\cosh(x)

Scope #

Recognizes Cosh

In

\mathrm{ExpToTrig}(\frac{1}{2}(e^{x}+\exp(-x)))

Out

\cosh(x)

Recognizes Sin

In

\mathrm{ExpToTrig}(\frac{\exp(i x)-\exp(-(i x))}{2i})

Out

\sin(x)

Real exponentials with base 3 are not rewritten

In

\mathrm{ExpToTrig}(3^{x}-3^{-x})

Out

3^{x}-3^{-x}

In

\mathrm{ExpToTrig}((-1)^{\frac{5}{7}})

Out

-1

In

\mathrm{ExpToTrig}(\bigl\lbrack\exp(i x)=-1, \sqrt[11]{-1}\bigr\rbrack)

Out

\bigl\lbrack i\sin(x)+\cos(x)=-1, -1\bigr\rbrack

Real logs are unchanged too

In

\mathrm{ExpToTrig}(\ln(x+2)-\ln(2-x))

Out

\ln(x+2)-\ln(2-x)

In

\mathrm{ExpToTrig}(e^{x}≤11∧3^{i x}=7)

Out

\sinh(x)+\cosh(x)≤11∧3^{i x}=7

In

\mathrm{ExpToTrig}(2^{i x})

Out

2^{i x}

In

\mathrm{ExpToTrig}(\ln(i x+1)-\ln(1-i x))

Out

\ln(i x+1)-\ln(1-i x)

Properties #

A root of unity stays symbolic

In

\mathrm{ExpToTrig}(\sqrt[16]{-1})

Out

\sqrt[16]{-1}

In

\mathrm{ExpToTrig}(\exp(i x)=-1)

Out

i\sin(x)+\cos(x)=-1

About

Examples	13
Checked against	Wolfram Language 6 agree7 differ
Crosswalk	Wolfram Language

See also: ComplexExpand, TrigToExp