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) andFE_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)
andFE_INVALID
is raised - If
z
is(x,NaN)
(for any positive finite x), the result is(NaN,NaN)
andFE_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) andFE_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)
andFE_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
#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
computes hyperbolic cosine of a complex number (ch(z)) (function template) | |
computes hyperbolic tangent of a complex number (function template) | |
(C++11) | computes area hyperbolic sine of a complex number (function template) |
computes hyperbolic sine (sh(x)) (function) | |
applies the function std::sinh to each element of valarray (function template) | |
C documentation for csinh |
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