std::sinh(std::complex) (3) - Linux Manuals

std::sinh(std::complex): std::sinh(std::complex)

NAME

std::sinh(std::complex) - std::sinh(std::complex)

Synopsis

Defined in header <complex>
template< class T > (since C++11)
complex<T> sinh( const complex<T>& z );

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 =

ez
-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)