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Cosh

Status: Stable

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

Description

Cosh[z]
    gives the hyperbolic cosine of z, (Exp[z] + Exp[-z]) / 2.
Cosh is Listable.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_cosh (src/hyperbolic.c) mirrors the trig cascade. (1) strip_inverse_call folds Cosh[ArcCosh[x]] -> x. (2) try_simp_forward_of_inverse_hyp handles Cosh[ArcSinh[x]] -> Sqrt[1+x^2] and Cosh[ArcTanh[x]] -> 1/Sqrt[1-x^2]. (3) even_fold for evenness Cosh[-x] -> Cosh[x]. (4) hyp_i_fold rewrites Cosh[I y] -> Cos[y]. (5) Cosh[0] -> 1; Cosh[±Infinity] -> Infinity. (6) Numeric fallback: MPFR via mpfr_cosh/mpfr_complex_cosh, else get_approx + C99 ccosh for inexact real/complex inputs. Otherwise NULL. There is no exact rational-multiple-of-Pi table for the hyperbolic heads.

Data structures. Expr* trees built with the shared make_* helpers.

Attributes: Listable, NumericFunction, 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]:= Cosh[0]
Out[1]= 1

In[2]:= Cosh[-x]
Out[2]= Cosh[x]

In[1]:= Cosh[Pi I]
Out[1]= -1

In[1]:= Cosh[ArcSinh[x]]
Out[1]= Sqrt[1 + x^2]

In[1]:= TrigExpand[Cosh[a + b]]
Out[1]= Cosh[a] Cosh[b] + Sinh[a] Sinh[b]

In[1]:= Series[Cosh[x], {x, 0, 6}]
Out[1]= 1 + 1/2 x^2 + 1/24 x^4 + 1/720 x^6 + O[x]^7

In[2]:= N[Cosh[1], 40]
Out[2]= 1.5430806348152437784779056207570616826015

Notes

Cosh[z] is the hyperbolic cosine, (Exp[z] + Exp[-z])/2. It is even, so the sign of the argument is dropped. Cosh is Listable.