# std::asin(std::complex)

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< cpp‎ | numeric‎ | complex

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

Computes complex arc sine of a complex value `z`. Branch cut exists outside the interval [−1 ; +1] along the real axis.

### Parameters

z | - | complex value

### Return value

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

Errors and special cases are handled as if the operation is implemented by `-i * std::asinh(i*z)`, where `i` is the imaginary unit.

### Notes

Inverse sine (or arc sine) is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-∞,-1) and (1,∞) of the real axis.

The mathematical definition of the principal value of arc sine is asin z = -iln(iz + √1-z2
)

For any z, asin(z) = acos(-z) -

π
2

### Example

Run this code

    #include <iostream>
    #include <cmath>
    #include <complex>

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

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

        // for any z, acos(z) = pi - acos(-z)
        const double pi = std::acos(-1);
        std::complex<double> z3 = pi - std::acos(z2);
        std::cout << "cos(pi - acos" << z2 << ") = " << std::cos(z3) << '\n';
    }

Output:

    asin(-2.000000,0.000000) = (-1.570796,1.316958)
    asin(-2.000000,-0.000000) (the other side of the cut) = (-1.570796,-1.316958)
    sin(acos(-2.000000,-0.000000) - pi/2) = (-2.000000,-0.000000)

### See also

acos(std::complex) (C++11) | computes arc cosine of a complex number (arccos(z)) (function template)
atan(std::complex) (C++11) | computes arc tangent of a complex number (arctan(z)) (function template)
sin(std::complex) | computes sine of a complex number (sin(z)) (function template)
asin | computes arc sine (arcsin(x)) (function)
asin(std::valarray) | applies the function std::asin to each element of valarray (function template)
C documentation for casin

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