TrigToExp

Trigonometric and hyperbolic functions rewritten exactly in terms of the complex exponential.

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

TrigToExp(expr)Extends compute-engine's TrigToExp with the logarithmic closed forms for Arcsin, Arctan, Arcoth, and Arcsch when one of these is the top-level operator; everything else is left to the native rewriter.

SageMathSR(expr).exponentialize()

Typed signatures

Type	Package	Applies
(value) -> value	analyticreplaces compute-engine	always
(value) -> value	compute-engine	always

Domain: Compute engine

Examples

Basic #

Cosine as the average of two complex exponentials

live ↗

In$\mathrm{TrigToExp}(\cos(x))$Out$\frac{1}{2}(e^{i x}+e^{-i x})$

Hyperbolic sine as a difference of real exponentials

live ↗

In$\mathrm{TrigToExp}(\sinh(x))$Out$\frac{1}{2}(e^{x}-e^{-x})$

Scope #

Tanh becomes a quotient of exponentials

live ↗

In$\mathrm{TrigToExp}(\tanh(x))$Out$\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

Arcsine becomes a logarithm: $-i\ln(ix+\sqrt{1-x^2})$

live ↗

In$\mathrm{TrigToExp}(\arcsin(x))$Out$-i\ln(i x+\sqrt{1-x^2})$

Inverse hyperbolic cotangent becomes a half-difference of logs

live ↗

In$\mathrm{TrigToExp}(\mathrm{arcoth}(x))$Out$\frac{1}{2}\ln(\frac{x+1}{x-1})$

Conversion reaches inside relations and conjunctions, leaving the arctan untouched

live ↗

In$\mathrm{TrigToExp}(\sinh(x)≤11∧\arctan(x)=7)$Out$\frac{1}{2}(e^{x}-e^{-x})≤11∧\arctan(x)=7$

live ↗

In$\mathrm{TrigToExp}(\bigl\lbrack\sin(x), \cos(x), \tan(x)\bigr\rbrack)$Out$\bigl\lbrack(-\frac{1}{2}i)e^{i x}+\frac{1}{2}ie^{-i x}, \frac{1}{2}(e^{i x}+e^{-i x}), \frac{-i(e^{i x}-e^{-i x})}{e^{i x}+e^{-i x}}\bigr\rbrack$

live ↗

In$\mathrm{TrigToExp}(\cos(x)+i\sin(x))$Out$\frac{1}{2}(e^{i x}+e^{-i x})+i((-\frac{1}{2}i)e^{i x}+\frac{1}{2}ie^{-i x})$

Applications #

Inner inverse trig stays inside the exponential

live ↗

In$\mathrm{TrigToExp}(\sin(5\arcsin(x)))$Out$(-\frac{1}{2}i)e^{5i\arcsin(x)}+\frac{1}{2}ie^{-5i\arcsin(x)}$

Properties #

Applies to every part of a chain of inequalities

live ↗

In$\mathrm{TrigToExp}(\sin(x)\lt x\lt\tan(x))$Out$(-\frac{1}{2}i)e^{i x}+\frac{1}{2}ie^{-i x}\lt x\lt\frac{-i(e^{i x}-e^{-i x})}{e^{i x}+e^{-i x}}$

The right-hand side $\pi/3$ is left as is

live ↗

In$\mathrm{TrigToExp}(\sin(x)=\frac{π}{3})$Out$(-\frac{1}{2}i)e^{i x}+\frac{1}{2}ie^{-i x}=\frac{π}{3}$

About

Examples	11
Checked against	Wolfram Language 11 agreeSageMath 8 agree1 unevaluatedSymPy 7 agree2 differ2 error
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedwolframpackages/oracle/src/mappings.ts

identity rename (see names.wolframIdentity); kernel-verified on the Arcsin/Arccoth closed forms this override adds too.

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts