std::sinh(std::complex)

 

C++

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Technical Specifications

 

Numerics library

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std::complex

Member functions
complex::complex
complex::operator=
complex::real
complex::imag
complex::operator+=complex::operator-=complex::operator*=complex::operator/=
Non-member functions
operator+operator- operator+operator-operator*operator/ operator==operator!= operator<<operator>>	real imag abs arg norm conj proj (C++11) polar operator""ioperator""ifoperator""il (C++14)(C++14)(C++14)
Exponential functions
exp
log
log10
Power functions
pow
sqrt
Trigonometric functions
sin cos tan	asin (C++11) acos (C++11) atan (C++11)
Hyperbolic functions
sinh cosh tanh	asinh (C++11) acosh (C++11) atanh (C++11)

 

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 =

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)

See also

cosh(std::complex)	computes hyperbolic cosine of a complex number (ch(z)) (function template)
tanh(std::complex)	computes hyperbolic tangent of a complex number (function template)
asinh(std::complex) (C++11)	computes area hyperbolic sine of a complex number (function template)
sinh	computes hyperbolic sine (sh(x)) (function)
sinh(std::valarray)	applies the function std::sinh to each element of valarray (function template)
C documentation for csinh