std::sinh(std::complex)
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std::complex
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(C++26)
Defined in header
<complex>
template< class T >
complex<T> sinh( const complex<T>& z );
(since C++11)
complex<T> sinh( const complex<T>& z );
Computes complex hyperbolic sine of a complex value z.
Contents
[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
-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]
(function template) [edit]
computes hyperbolic tangent of a complex number (\({\small\tanh{z}}\)tanh(z))
(function template) [edit]
(function template) [edit]
(C++11)
(function template) [edit]
C documentation for csinh