cacosf, cacos, cacosl

From cppreference.com
< clrm; | numericlrm; | complex
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 =
1
2
+ iln(iz + 1-z2
)

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

Example

#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)

See also

(C99)(C99)(C99)
computes the complex arc sine
(function)
(C99)(C99)(C99)
computes the complex arc tangent
(function)
(C99)(C99)(C99)
computes the complex cosine
(function)
(C99)(C99)
computes arc cosine (arccos(x))
(function)