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cacosf, cacos, cacosl

From cppreference.com
< c‎ | numeric‎ | 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.

Contents

[edit] Parameters

z - complex argument

[edit] 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 [−iπ ; iπ] along the imaginary axis.

[edit] 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 π/2-0i
  • If z is ±0+NaNi, the result is π/2+NaNi
  • If z is x+∞i (for any finite x), the result is π/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 π-∞i
  • If z is -∞+yi (for any positive finite y), the result is +0-∞i
  • If z is -∞+∞i, the result is 3π/4-∞i
  • If z is +∞+∞i, the result is π/4-∞i
  • If z is ±∞+NaNi, the result is NaN±∞i (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

[edit] 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
π + iln(iz + 1-z2
)

For any z, acos(z) = π - acos(-z)

[edit] 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

[edit] See also

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