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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.

Notes Sinh is Listable.

Examples (7)

Every input below was run against the current Mathilda build and its output recorded.

Applications (7)

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[5]:= Sinh[I x]
Out[5]= I Sin[x]

In[6]:= TrigExpand[Sinh[x + y]]
Out[6]= Cosh[x] Sinh[y] + Sinh[x] Cosh[y]

In[7]:= N[Sinh[1], 40]
Out[7]= 1.1752011936438014568823818505956008151557

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.

References

Notes & additional examples

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.