Sinh¶

Status: Stable

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

Description¶

Sinh[z]
    gives the hyperbolic sine of z, (Exp[z] - Exp[-z]) / 2.
Sinh is Listable.

Examples¶

No verified examples yet for this function.

Implementation notes¶

Algorithm. builtin_sinh is the hyperbolic analogue of builtin_sin: strip_inverse_call(arg, "ArcSinh") for Sinh[ArcSinh[x]] -> x; try_simp_forward_of_inverse_hyp for Sinh of the other inverse hyperbolics (Sinh[ArcCosh[x]] -> Sqrt[x-1] Sqrt[x+1], Sinh[ArcTanh[x]] -> x/Sqrt[1-x^2]); odd_fold for Sinh[-x] -> -Sinh[x]; hyp_i_fold(arg, "Sin", +1) for Sinh[I y] -> I Sin[y]. Special points: Sinh[0] = 0, Sinh[Infinity] = Infinity, Sinh[-Infinity] = -Infinity.

Numeric. MPFR values evaluate via numeric_mpfr_apply_unary(..., mpfr_sinh) with an mpfr_complex_sinh complex fallback; otherwise get_approx + csinh yields a real or Complex result for inexact arguments. 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¶

Source: src/hyperbolic.c
Specification: docs/spec/builtins/elementary-functions.md

Notes & additional examples¶

Worked examples¶

In[1]:= Sinh[0]
Out[1]= 0

In[2]:= N[Sinh[1]]
Out[2]= 1.1752

In[3]:= Sinh[-x]
Out[3]= -Sinh[x]

In[4]:= Sinh[ArcSinh[x]]
Out[4]= x

In[1]:= Sinh[I x]
Out[1]= I Sin[x]

In[2]:= TrigExpand[Sinh[x + y]]
Out[2]= Cosh[x] Sinh[y] + Sinh[x] Cosh[y]

In[3]:= N[Sinh[1], 40]
Out[3]= 1.1752011936438014568823818505956008151557

Notes¶

Sinh[z] is the hyperbolic sine, (Exp[z] - Exp[-z])/2. It is odd, so negative arguments pull the sign out front. Sinh is Listable.