# casinf, casin, casinl

< c‎ | numeric‎ | complex

C
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Complex number arithmetic
Types and the imaginary constant
Manipulation
Power and exponential functions
Trigonometric functions
 cacos casin catan
Hyperbolic functions

 Defined in header  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  #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 asin(iy) = i asinh(y), and the return type of the macro is imaginary.

## Contents

### Parameters

 z - 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 asin z = -iln(iz + 1-z2
)

For any z, asin(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)