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ExpToTrig

Status: Stable

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

Description

ExpToTrig[expr]
    rewrites exponentials and logarithms in expr in terms of circular and
    hyperbolic trigonometric functions when possible.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= ExpToTrig[Exp[I x]]
Out[1]= Cos[x] + I Sin[x]

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

In[3]:= ExpToTrig[Exp[I x] == -1]
Out[3]= Cos[x] + I Sin[x] == -1

Implementation notes

Algorithm. builtin_exptotrig is the partial inverse of TrigToExp. It runs a four-stage pipeline on the argument: (1) ReplaceRepeated with exp_to_trig_rules — E^(I x) :> Cos[x] + I Sin[x], E^(-x) :> Cosh[x] - Sinh[x], E^x :> Cosh[x] + Sinh[x], plus Log-combination patterns that fold back to ArcTan/ArcTanh/ArcSin/ArcCsch/... ; (2) Together to combine over a common denominator; (3) Cancel to reduce the rational; (4) a final ReplaceRepeated with exp_to_trig_simp that recovers reciprocal heads (Sin/Cos :> Tan, Cos^-1 :> Sec, etc.). The trig canonicalizer is suppressed across the whole pipeline (trig_canon_suppress_inc/dec) so the intermediate Sin/Cos forms survive long enough for Together/Cancel to act.

Data structures. Three static rule lists (exp_to_trig_rules, exp_to_trig_simp, plus the shared trig_factor_* machinery) parsed once in trigsimp_init. Results are routed through the active FactorMemo by the public builtin_exptotrig wrapper path via trig_memo_call semantics shared with the other trig builtins.

  • Listable, Protected.
  • Tries when possible to give results that do not involve explicit complex numbers.
  • ExpToTrig is natively the inverse of TrigToExp.
  • 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]:= ExpToTrig[Exp[x]]
Out[1]= Cosh[x] + Sinh[x]

In[1]:= ExpToTrig[Exp[I x]]
Out[1]= Cos[x] + I Sin[x]

In[1]:= ExpToTrig[(E^x - E^(-x))/2]
Out[1]= Sinh[x]

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

Notes

ExpToTrig rewrites exponentials in terms of circular and hyperbolic functions: a real exponent yields Cosh + Sinh, an imaginary one yields Cos + I Sin. It recognises the classical combinations, folding the half-difference (E^x - E^(-x))/2 back to Sinh[x] and the sum E^(I x) + E^(-I x) to 2 Cos[x].