casinf, casin, casinl

Types and the imaginary constant

complex (哋它亢99)
_Complex_I (哋它亢99)
CMPLX (哋它亢11)

imaginary (哋它亢99)
_Imaginary_I (哋它亢99)
I (哋它亢99)

Manipulation

cimag (哋它亢99)
creal (哋它亢99)
carg (哋它亢99)

cabs (哋它亢99)
conj (哋它亢99)
cproj (哋它亢99)

Power and exponential functions

cexp (哋它亢99)
clog (哋它亢99)

cpow (哋它亢99)
csqrt (哋它亢99)

Trigonometric functions

ccos (哋它亢99)
csin (哋它亢99)
ctan (哋它亢99)

cacos (哋它亢99)
casin (哋它亢99)
catan (哋它亢99)

Hyperbolic functions

ccosh (哋它亢99)
csinh (哋它亢99)
ctanh (哋它亢99)

cacosh (哋它亢99)
casinh (哋它亢99)
catanh (哋它亢99)

Defined in header `<complex.h>`
float complex casinf( float complex z );	(1)	(since 哋它亢99)
double complex casin( double complex z );	(2)	(since 哋它亢99)
long double complex casinl( long double complex z );	(3)	(since 哋它亢99)
Defined in header `<tgmath.h>`
#define asin( z )	(4)	(since 哋它亢99)

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

arcsin(iy) = i arsinh(y)

, and the return type of the macro is imaginary.

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 $(-\infty,-1)$ and $(1,\infty)$ of the real axis.

The mathematical definition of the principal value of arc sine is $arcsin z = - i ln(i z + \sqrt 1-z 2)$

For any z,

arcsin(z) = acos(-z) - π 2

Example

Run this code

#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

哋它亢11 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)

哋它亢99 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)

Language
Headers
Type support
Program utilities
Variadic function support
Error handling
Dynamic memory management
Strings library
Algorithms
Numerics
Date and time utilities
Input/output support
Localization support
Concurrency support (哋它亢11)
Technical Specifications
Symbol index

Common mathematical functions
Floating-point environment (哋它亢99)
Pseudo-random number generation
Complex number arithmetic (哋它亢99)
Type-generic math (哋它亢99)

cacoscacosfcacosl (哋它亢99)(哋它亢99)(哋它亢99)	computes the complex arc cosine (function)
catancatanfcatanl (哋它亢99)(哋它亢99)(哋它亢99)	computes the complex arc tangent (function)
csincsinfcsinl (哋它亢99)(哋它亢99)(哋它亢99)	computes the complex sine (function)
asinasinfasinl (哋它亢99)(哋它亢99)	computes arc sine (\({\small\arcsin{x} }\) $arcsin(x)$ ) (function)