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std::sinh(std::complex)

From cppreference.com
< cpp‎ | numeric‎ | complex
 
 
Numerics library
 
 
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.

[edit] Parameters

z - complex value

[edit] Return value

If no errors occur, complex hyperbolic sine of z is returned.

[edit] 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).

[edit] 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.

[edit] Example

Run this code
#include <cmath>
#include <complex>
#include <iostream>
 
int main()
{
 std::cout << std::fixed ;
 std::complex <double> z(1.0, 0.0); // 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(0.0, 1.0); // 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)

[edit] See also

computes hyperbolic cosine of a complex number (\({\small\cosh{z}}\)cosh(z))
(function template) [edit]
computes hyperbolic tangent of a complex number (\({\small\tanh{z}}\)tanh(z))
(function template) [edit]
computes area hyperbolic sine of a complex number (\({\small\operatorname{arsinh}{z}}\)arsinh(z))
(function template) [edit]
(C++11)(C++11)
computes hyperbolic sine (\({\small\sinh{x}}\)sinh(x))
(function) [edit]
applies the function std::sinh to each element of valarray
(function template) [edit]
C documentation for csinh
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