casinf, casin, casinl

 

C

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Technical Specifications

Symbol index

 

Numerics

Common mathematical functions

Floating-point environment (C99)

Pseudo-random number generation

Complex number arithmetic (C99)

Type-generic math (C99)

 

Complex number arithmetic

Types and the imaginary constant
complex (C99) _Complex_I (C99) CMPLX (C11)	imaginary (C99) _Imaginary_I (C99) I (C99)
Manipulation
cimag (C99) creal (C99) carg (C99)	cabs (C99) conj (C99) cproj (C99)
Power and exponential functions
cexp (C99) clog (C99)	cpow (C99) csqrt (C99)
Trigonometric functions
ccos (C99) csin (C99) ctan (C99)	cacos (C99) casin (C99) catan (C99)
Hyperbolic functions
ccosh (C99) csinh (C99) ctanh (C99)	cacosh (C99) casinh (C99) catanh (C99)

 

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

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 \(\small \arcsin z = -{\rm i}\ln({\rm i}z+\sqrt{1-z^2})\)arcsin z = -iln(iz + √1-z2
)

For any z, \(\small{ \arcsin(z) = \arccos(-z) - \frac{\pi}{2} }\)arcsin(z) = acos(-z) -

π

2

Example

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

• 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)

See also

cacoscacosfcacosl (C99)(C99)(C99)	computes the complex arc cosine (function)
catancatanfcatanl (C99)(C99)(C99)	computes the complex arc tangent (function)
csincsinfcsinl (C99)(C99)(C99)	computes the complex sine (function)
asinasinfasinl (C99)(C99)	computes arc sine (\({\small\arcsin{x} }\)arcsin(x)) (function)