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TrigToExp

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

TrigToExp[expr]
    rewrites circular and hyperbolic trigonometric functions (and their
    inverses) in expr in terms of exponentials and logarithms.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= TrigToExp[Cos[x]]
Out[1]= 1/2 E^(-I x) + 1/2 E^(I x)

In[2]:= TrigToExp[ArcSin[x]]
Out[2]= -I Log[I x + Sqrt[1 - x^2]]

In[3]:= TrigToExp[{Sin[x], Cos[x], Tan[x]}]
Out[3]= {(-1/2*I) E^(I x) + (1/2*I) E^(-I x), 1/2 E^(-I x) + 1/2 E^(I x), -I E^(I x)/(E^(-I x) + E^(I x)) + I E^(-I x)/(E^(-I x) + E^(I x))}

Implementation notes

Algorithm. builtin_trigtoexp rewrites circular, hyperbolic, and their inverse functions into complex-exponential / logarithmic form. It applies the trig_to_exp_rules rule list (a ReplaceAll) — e.g. Sin[x] :> I/2 E^(-I x) - I/2 E^(I x), Cos[x] :> (E^(-I x) + E^(I x))/2, Tan/Cot/Sec/Csc, the Sinh/Cosh/Tanh/... hyperbolic analogues, and the ArcSin :> -I Log[...] family of inverse identities — then runs Expand to distribute the result into a flat sum. The Times/Power-level trig canonicalizer is suppressed for the duration (trig_canon_suppress_inc/dec) so the intermediate Sin/Cos forms are not prematurely re-collapsed back to Tan etc. before the rule fires.

Data structures. The rules are a single parse_expression'd List of RuleDelayeds, built once in trigsimp_init and stored in the static trig_to_exp_rules. Results are memoized through the active FactorMemo (keyed under a TrigToExp[arg] head) so repeated Simplify candidate-set calls on identical subexpressions hit the cache instead of re-running ReplaceAll + Expand, which costs 30–130 ms on Tan-rich inputs.

  • Listable, Protected.
  • Operates on both circular and hyperbolic functions, and their inverses.
  • Automatically threads over lists, equations, inequalities, and logic functions.

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= TrigToExp[Sin[x]]
Out[1]= (-1/2*I) E^(I x) + (1/2*I) E^(-I x)

In[1]:= TrigToExp[Cos[x]]
Out[1]= 1/2 E^(-I x) + 1/2 E^(I x)

Hyperbolic functions become real exponentials:

In[1]:= TrigToExp[Cosh[x]]
Out[1]= 1/2 E^x + 1/2 E^(-x)

The inverse functions are rewritten via their logarithmic forms — here the classic arctan identity:

In[1]:= TrigToExp[ArcTan[x]]
Out[1]= (-1/2*I) Log[1 + I x] + (1/2*I) Log[1 - I x]

In[1]:= TrigToExp[ArcSin[x]]
Out[1]= -I Log[I x + Sqrt[1 - x^2]]