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catanf, catan, catanl

Defined in header `<complex.h>`
float complex catanf( float complex z );	(1)	(since C99)
double complex catan( double complex z );	(2)	(since C99)
long double complex catanl( long double complex z );	(3)	(since C99)
Defined in header `<tgmath.h>`
#define atan( z )	(4)	(since C99)

1-3) Computes the complex arc tangent of z with branch cuts outside the interval [−i,+i] along the imaginary axis.

4) Type-generic macro: If z has type long double complex, catanl is called. if z has type double complex, catan is called, if z has type float complex, catanf is called. If z is real or integer, then the macro invokes the corresponding real function (atanf, atan, atanl). If z is imaginary, then the macro invokes the corresponding real version of the function atanh, implementing the formula atan(iy) = i atanh(y), and the return type of the macro is imaginary.

Parameters

complex argument

Return value

If no errors occur, complex arc tangent 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 * catanh(I*z).

Notes

Inverse tangent (or arc tangent) is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-∞i,-i) and (+i,+∞i) of the imaginary axis. The mathematical definition of the principal value of inverse tangent is atan z = -

1/2 i [ln(1 - iz) - ln (1 + iz]

Example

#include <stdio.h>
#include <float.h>
#include <complex.h>
 
int main(void)
{
    double complex z = catan(2*I);
    printf("catan(+0+2i) = %f%+fi\n", creal(z), cimag(z));
 
    double complex z2 = catan(-conj(2*I)); // or CMPLX(-0.0, 2)
    printf("catan(-0+2i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
 
    double complex z3 = 2*catan(2*I*DBL_MAX); // or CMPLX(0, INFINITY)
    printf("2*catan(+0+i*Inf) = %f%+fi\n", creal(z3), cimag(z3));
}

Output:

catan(+0+2i) = 1.570796+0.549306i
catan(-0+2i) (the other side of the cut) = -1.570796+0.549306i
2*catan(+0+i*Inf) = 3.141593+0.000000i

References

C11 standard (ISO/IEC 9899:2011):

7.3.5.3 The catan functions (p: 191)
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):

7.3.5.3 The catan functions (p: 173)
7.22 Type-generic math <tgmath.h> (p: 335-337)
G.7 Type-generic math <tgmath.h> (p: 480)