std::sinh(std::complex)

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 floating-point 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 = e z -e -z 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)

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