/C++

# std::asinh(std::complex)

Defined in header `<complex>`
```template< class T >
complex<T> asinh( const complex<T>& z );```
(since C++11)

Computes complex arc hyperbolic sine of a complex value `z` with branch cuts outside the interval [−i; +i] along the imaginary axis.

### Parameters

 z - complex value

### Return value

If no errors occur, the complex arc hyperbolic sine of `z` is returned, in the range of a strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.

### Error handling and special values

Errors are reported consistent with `math_errhandling`.

If the implementation supports IEEE floating-point arithmetic,

• `std::asinh(std::conj(z)) == std::conj(std::asinh(z))`
• `std::asinh(-z) == -std::asinh(z)`
• If `z` is `(+0,+0)`, the result is `(+0,+0)`
• If `z` is `(x,+∞)` (for any positive finite x), the result is `(+∞,π/2)`
• If `z` is `(x,NaN)` (for any finite x), the result is `(NaN,NaN)` and `FE_INVALID` may be raised
• If `z` is `(+∞,y)` (for any positive finite y), the result is `(+∞,+0)`
• If `z` is `(+∞,+∞)`, the result is `(+∞,π/4)`
• If `z` is `(+∞,NaN)`, the result is `(+∞,NaN)`
• 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,+∞)`, the result is `(±∞,NaN)` (the sign of the real part is unspecified)
• If `z` is `(NaN,NaN)`, the result is `(NaN,NaN)`

Although the C++ standard names this function "complex arc hyperbolic sine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic sine", and, less common, "complex area hyperbolic sine".

Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-i∞,-i) and (i,i∞) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + 1+z2
) For any z, asinh(z) =

 asin(iz) i

### Example

```#include <iostream>
#include <complex>

int main()
{
std::cout << std::fixed;
std::complex<double> z1(0, -2);
std::cout << "asinh" << z1 << " = " << std::asinh(z1) << '\n';

std::complex<double> z2(-0.0, -2);
std::cout << "asinh" << z2 << " (the other side of the cut) = "
<< std::asinh(z2) << '\n';

// for any z, asinh(z) = asin(iz)/i
std::complex<double> z3(1,2);
std::complex<double> i(0,1);
std::cout << "asinh" << z3 << " = " << std::asinh(z3) << '\n'
<< "asin" << z3*i << "/i = " << std::asin(z3*i)/i << '\n';
}```

Output:

```asinh(0.000000,-2.000000) = (1.316958,-1.570796)
asinh(-0.000000,-2.000000) (the other side of the cut) = (-1.316958,-1.570796)
asinh(1.000000,2.000000) = (1.469352,1.063440)
asin(-2.000000,1.000000)/i = (1.469352,1.063440)```