std::sinh(std::complex)
From cppreference.com
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
.
Parameters
z    complex value 
Return value
If no errors occur, complex hyperbolic sine of z
is returned
Error handling and special values
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)
 If
z
is(+0,+0)
, the result is(+0,+0)
 If
z
is(+0,+∞)
, the result is(±0,NaN)
(the sign of the real part is unspecified) and FE_INVALID is raised  If
z
is(+0,NaN)
, the result is(±0,NaN)
 If
z
is(x,+∞)
(for any positive finite x), the result is(NaN,NaN)
and FE_INVALID is raised  If
z
is(x,NaN)
(for any positive finite x), the result is(NaN,NaN)
and FE_INVALID may be raised  If
z
is(+∞,+0)
, the result is(+∞,+0)
 If
z
is(+∞,y)
(for any positive finite y), the result is(+∞,cis(y))
 If
z
is(+∞,+∞)
, the result is(±∞,NaN)
(the sign of the real part is unspecified) and FE_INVALID is raised  If
z
is(+∞,NaN)
, the result is(±∞,NaN)
(the sign of the real part is unspecified)  If
z
is(NaN,+0)
, the result is(NaN,+0)
 If
z
is(NaN,y)
(for any finite nonzero y), the result is(NaN,NaN)
and FE_INVALID may be raised  If
z
is(NaN,NaN)
, the result is(NaN,NaN)
where cis(y) is cos(y) + i sin(y)
Notes
Mathematical definition of hyperbolic sine is sinh z =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
Example
Run this code
#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)
See also
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
