Defined in header <complex>  

template< class T > complex<T> sinh( const complex<T>& z );  (since C++11) 
Computes complex hyperbolic sine of a complex value z
.
z    complex value 
If no errors occur, complex hyperbolic sine of z
is returned.
Errors are reported consistent with math_errhandling
.
If the implementation supports IEEE floatingpoint arithmetic,
std::sinh(std::conj(z)) == std::conj(std::sinh(z))
std::sinh(z) == std::sinh(z)
z
is (+0,+0)
, the result is (+0,+0)
z
is (+0,+∞)
, the result is (±0,NaN)
(the sign of the real part is unspecified) and FE_INVALID
is raised z
is (+0,NaN)
, the result is (±0,NaN)
z
is (x,+∞)
(for any positive finite x), the result is (NaN,NaN)
and FE_INVALID
is raised z
is (x,NaN)
(for any positive finite x), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (+∞,+0)
, the result is (+∞,+0)
z
is (+∞,y)
(for any positive finite y), the result is (+∞,cis(y))
z
is (+∞,+∞)
, the result is (±∞,NaN)
(the sign of the real part is unspecified) and FE_INVALID
is raised z
is (+∞,NaN)
, the result is (±∞,NaN)
(the sign of the real part is unspecified) z
is (NaN,+0)
, the result is (NaN,+0)
z
is (NaN,y)
(for any finite nonzero y), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (NaN,NaN)
, the result is (NaN,NaN)
where cis(y) is cos(y) + i sin(y).
ez ez 
2 
Hyperbolic sine is an entire function in the complex plane and has no branch cuts. It is periodic with respect to the imaginary component, with period 2πi.
#include <iostream> #include <cmath> #include <complex> int main() { std::cout << std::fixed; std::complex<double> z(1, 0); // behaves like real sinh along the real line std::cout << "sinh" << z << " = " << std::sinh(z) << " (sinh(1) = " << std::sinh(1) << ")\n"; std::complex<double> z2(0, 1); // behaves like sine along the imaginary line std::cout << "sinh" << z2 << " = " << std::sinh(z2) << " ( sin(1) = " << std::sin(1) << ")\n"; }
Output:
sinh(1.000000,0.000000) = (1.175201,0.000000) (sinh(1) = 1.175201) sinh(0.000000,1.000000) = (0.000000,0.841471) ( sin(1) = 0.841471)
computes hyperbolic cosine of a complex number (ch(z)) (function template) 

computes hyperbolic tangent of a complex number (function template) 

(C++11)  computes area hyperbolic sine of a complex number (function template) 
computes hyperbolic sine (sh(x)) (function) 

applies the function std::sinh to each element of valarray (function template) 

C documentation for csinh 
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