# std::sinh(std::complex)

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< cpp‎ | numeric‎ | complex

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 <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)

### See also

cosh(std::complex) | computes hyperbolic cosine of a complex number (\({\small\cosh{z} }\)cosh(z)) (function template)
tanh(std::complex) | computes hyperbolic tangent of a complex number (\({\small\tanh{z} }\)tanh(z)) (function template)
asinh(std::complex) (C++11) | computes area hyperbolic sine of a complex number (\({\small\operatorname{arsinh}{z} }\)arsinh(z)) (function template)
sinh sinhf sinhl (C++11)(C++11) | computes hyperbolic sine (\({\small\sinh{x} }\)sinh(x)) (function)
sinh(std::valarray) | applies the function std::sinh to each element of valarray (function template)

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