cacosf, cacos, cacosl

 

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Complex number arithmetic

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complex _Complex_I CMPLX	imaginary _Imaginary_I I
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cimag creal carg	cabs conj cproj
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cexp clog	cpow csqrt
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ccos csin ctan	cacos casin catan
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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 [−iπ ; iπ] 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 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

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)

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

See also

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)
C++ documentation for acos