casinf, casin, casinl

Defined in header `<complex.h>`
float complex casinf( float complex z );	(1)	(since C99)
double complex casin( double complex z );	(2)	(since C99)
long double complex casinl( long double complex z );	(3)	(since C99)
Defined in header `<tgmath.h>`
#define asin( z )	(4)	(since C99)

1-3) Computes the complex arc sine of z with branch cuts outside the interval [−1,+1] along the real axis.

4) Type-generic macro: If z has type long double complex, casinl is called. if z has type double complex, casin is called, if z has type float complex, casinf is called. If z is real or integer, then the macro invokes the corresponding real function (asinf, asin, asinl). If z is imaginary, then the macro invokes the corresponding real version of the function asinh, implementing the formula \(\small \arcsin({\rm i}y) = {\rm i}{\rm arsinh}(y)\)arcsin(iy) = i arsinh(y), and the return type of the macro is imaginary.

Parameters

complex argument

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 * casinh(I*z)

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 \(\small \arcsin z = -{\rm i}\ln({\rm i}z+\sqrt{1-z^2})\)arcsin z = -iln(iz + √1-z²) For any z, \(\small{ \arcsin(z) = \arccos(-z) - \frac{\pi}{2} }\)arcsin(z) = acos(-z) -

π/2

Example

#include <stdio.h>
#include <math.h>
#include <complex.h>
 
int main(void)
{
    double complex z = casin(-2);
    printf("casin(-2+0i) = %f%+fi\n", creal(z), cimag(z));
 
    double complex z2 = casin(conj(-2)); // or CMPLX(-2, -0.0)
    printf("casin(-2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
 
    // for any z, asin(z) = acos(-z) - pi/2
    double pi = acos(-1);
    double complex z3 = csin(cacos(conj(-2))-pi/2);
    printf("csin(cacos(-2-0i)-pi/2) = %f%+fi\n", creal(z3), cimag(z3));
}

Output:

casin(-2+0i) = -1.570796+1.316958i
casin(-2-0i) (the other side of the cut) = -1.570796-1.316958i
csin(cacos(-2-0i)-pi/2) = 2.000000+0.000000i

References

C11 standard (ISO/IEC 9899:2011):

7.3.5.2 The casin functions (p: 190)
7.25 Type-generic math <tgmath.h> (p: 373-375)
G.7 Type-generic math <tgmath.h> (p: 545)

C99 standard (ISO/IEC 9899:1999):

7.3.5.2 The casin functions (p: 172)
7.22 Type-generic math <tgmath.h> (p: 335-337)
G.7 Type-generic math <tgmath.h> (p: 480)