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Tanh

Status: Stable

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

Description

Tanh[z]
    gives the hyperbolic tangent of z, Sinh[z] / Cosh[z].
Tanh is Listable.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_tanh follows the same hyperbolic cascade in src/hyperbolic.c: strip_inverse_call(arg, "ArcTanh") for Tanh[ArcTanh[x]] -> x; try_simp_forward_of_inverse_hyp for Tanh of the other inverse hyperbolics (Tanh[ArcSinh[x]] -> x/Sqrt[1+x^2], Tanh[ArcCosh[x]] -> Sqrt[x-1] Sqrt[x+1]/x, Tanh[ArcCoth[x]] -> 1/x); odd_fold for Tanh[-x] -> -Tanh[x]; hyp_i_fold(arg, "Tan", +1) for Tanh[I y] -> I Tan[y]. Special points: Tanh[0] = 0, Tanh[Infinity] = 1, Tanh[-Infinity] = -1.

Numeric. MPFR values use numeric_mpfr_apply_unary(..., mpfr_tanh) (complex fallback mpfr_complex_tanh); otherwise get_approx + ctanh covers inexact real/complex inputs. Symbolic input returns NULL. Attributes: ATTR_LISTABLE | ATTR_NUMERICFUNCTION | ATTR_PROTECTED.

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]:= Tanh[0]
Out[1]= 0

In[2]:= N[Tanh[1]]
Out[2]= 0.761594

In[3]:= Tanh[ArcTanh[z]]
Out[3]= z

In[1]:= N[Tanh[1], 40]
Out[1]= 0.76159415595576488811945828260479359041279

In[1]:= D[Tanh[x], x]
Out[1]= Sech[x]^2

In[1]:= Series[Tanh[x], {x, 0, 7}]
Out[1]= x - 1/3 x^3 + 2/15 x^5 - 17/315 x^7 + O[x]^8

Notes

Tanh[z] is the hyperbolic tangent, Sinh[z]/Cosh[z]. Tanh is Listable.