cacosf, cacos, cacosl

Defined in header `<complex.h>`
float complex cacosf( float complex z );	(1)	(since C99)
double complex cacos( double complex z );	(2)	(since C99)
long double complex cacosl( long double complex z );	(3)	(since C99)
Defined in header `<tgmath.h>`
#define acos( z )	(4)	(since C99)

1-3) Computes the complex arc cosine 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, cacosl is called. if z has type double complex, cacos is called, if z has type float complex, cacosf is called. If z is real or integer, then the macro invokes the corresponding real function (acosf, acos, acosl). If z is imaginary, then the macro invokes the corresponding complex number version.

Parameters

z

-

complex argument

Return value

If no errors occur, complex arc cosine of z is returned, in the range $[0; )$ along the real axis and in the range $[i; i]$ along the imaginary axis.

Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

cacos(conj(z)) == conj(cacos(z))
If z is 0+0i, the result is /2-0i
If z is 0+NaNi, the result is /2+NaNi
If z is x+i (for any finite x), the result is /2-i
If z is x+NaNi (for any nonzero finite x), the result is NaN+NaNi and FE_INVALID may be raised.
If z is -+yi (for any positive finite y), the result is -i
If z is -+yi (for any positive finite y), the result is +0-i
If z is -+i, the result is 3/4-i
If z is ++i, the result is /4-i
If z is +NaNi, the result is NaNi (the sign of the imaginary part is unspecified)
If z is NaN+yi (for any finite y), the result is NaN+NaNi and FE_INVALID may be raised
If z is NaN+i, the result is NaN-i
If z is NaN+NaNi, the result is NaN+NaNi

Notes

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

The mathematical definition of the principal value of arc cosine is

acos z = 12 + i ln(i z + 1-z 2)

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

Example

Run this code

#include <stdio.h>
#include <math.h>
#include <complex.h>

int main(void)
{
    double complex z = cacos(-2);
    printf("cacos(-2+0i) =%f%+fi\n", creal(z), cimag(z));

    double complex z2 = cacos(conj(-2)); // or CMPLX(-2, -0.0)
    printf("cacos(-2-0i) (the other side of the cut) =%f%+fi\n", creal(z2), cimag(z2));

    // for any z, acos(z) = pi - acos(-z)
    double pi = acos(-1);
    double complex z3 = ccos(pi-z2);
    printf("ccos(pi - cacos(-2-0i) =%f%+fi\n", creal(z3), cimag(z3));
}

Output:

cacos(-2+0i) = 3.141593-1.316958i
cacos(-2-0i) (the other side of the cut) = 3.141593+1.316958i
ccos(pi - cacos(-2-0i) = 2.000000+0.000000i

References

C11 standard (ISO/IEC 9899:2011):

7.3.5.1 The cacos functions (p: 190)

7.25 Type-generic math <tgmath.h> (p: 373-375)

G.6.1.1 The cacos functions (p: 539)

G.7 Type-generic math <tgmath.h> (p: 545)

C99 standard (ISO/IEC 9899:1999):

7.3.5.1 The cacos functions (p: 172)

7.22 Type-generic math <tgmath.h> (p: 335-337)

G.6.1.1 The cacos functions (p: 474)

G.7 Type-generic math <tgmath.h> (p: 480)

Language
Headers
Type support
Program utilities
Variadic function support
Error handling
Dynamic memory management
Date and time utilities
Strings library
Algorithms
Numerics
Input/output support
Localization support
Atomic operations (C11)
Thread support (C11)
Technical Specifications

Common mathematical functions
Floating-point environment
Complex number arithmetics
Random number generation
Type-generic math

casincasinfcasinl (C99)(C99)(C99)	computes the complex arc sine (function)
catancatanfcatanl (C99)(C99)(C99)	computes the complex arc tangent (function)
ccosccosfccosl (C99)(C99)(C99)	computes the complex cosine (function)
acosacosfacosl (C99)(C99)	computes arc cosine ( $arccos(x)$ ) (function)