ExpToTrig

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

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

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

live ↗

In$\mathrm{ExpToTrig}(e^{i x})$Out$i\sin(x)+\cos(x)$

A real exponential splits into hyperbolic parts

live ↗

In$\mathrm{ExpToTrig}(e^{x})$Out$\sinh(x)+\cosh(x)$

Scope #

Recognizes Cosh

live ↗

In$\mathrm{ExpToTrig}(\frac{1}{2}(e^{x}+e^{-x}))$Out$\cosh(x)$

Recognizes Sin

live ↗

In$\mathrm{ExpToTrig}(\frac{e^{i x}-e^{-(i x)}}{2i})$Out$\sin(x)$

Real exponentials with base 3 are not rewritten

live ↗

In$\mathrm{ExpToTrig}(3^{x}-3^{-x})$Out$3^{x}-3^{-x}$

live ↗

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

live ↗

In$\mathrm{ExpToTrig}(\bigl\lbrack e^{i x}=-1, \sqrt[11]{-1}\bigr\rbrack)$Out$\bigl\lbrack i\sin(x)+\cos(x)=-1, -1\bigr\rbrack$

Real logs are unchanged too

live ↗

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

live ↗

In$\mathrm{ExpToTrig}(e^{x}≤11∧3^{i x}=7)$Out$\sinh(x)+\cosh(x)≤11∧3^{i x}=7$

live ↗

In$\mathrm{ExpToTrig}(2^{i x})$Out$2^{i x}$

live ↗

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

live ↗

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

live ↗

In$\mathrm{ExpToTrig}(e^{i x}=-1)$Out$i\sin(x)+\cos(x)=-1$

About

Examples	13
Checked against	Wolfram Language 6 agree7 differSymPy 5 agree5 differ1 unevaluated2 error
Crosswalk	Wolfram Language

Implementation

mappedsympypackages/oracle/src/mappings.ts

See also: ComplexExpand, TrigToExp