/C

# clogf, clog, clogl

Defined in header `<complex.h>`
`float complex       clogf( float complex z );`
(1) (since C99)
`double complex      clog( double complex z );`
(2) (since C99)
`long double complex clogl( long double complex z );`
(3) (since C99)
Defined in header `<tgmath.h>`
`#define log( z )`
(4) (since C99)
1-3) Computes the complex natural (base-e) logarithm of `z` with branch cut along the negative real axis.
4) Type-generic macro: If `z` has type `long double complex`, `clogl` is called. if `z` has type `double complex`, `clog` is called, if `z` has type `float complex`, `clogf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (`logf`, `log`, `logl`). If `z` is imaginary, the corresponding complex number version is called.

### Parameters

 z - complex argument

### Return value

If no errors occur, the complex natural logarithm of `z` is returned, in the range of a strip in the interval [−iπ, +iπ] along the imaginary axis and mathematically unbounded along the real axis.

### Error handling and special values

Errors are reported consistent with math_errhandling.

If the implementation supports IEEE floating-point arithmetic,

• The function is continuous onto the branch cut taking into account the sign of imaginary part
• `clog(conj(z)) == conj(clog(z))`
• If `z` is `-0+0i`, the result is `-∞+πi` and `FE_DIVBYZERO` is raised
• If `z` is `+0+0i`, the result is `-∞+0i` and `FE_DIVBYZERO` is raised
• If `z` is `x+∞i` (for any finite x), the result is `+∞+πi/2`
• If `z` is `x+NaNi` (for any finite x), the result is `NaN+NaNi` and `FE_INVALID` may be raised
• If `z` is `-∞+yi` (for any finite positive y), the result is `+∞+πi`
• If `z` is `+∞+yi` (for any finite positive y), the result is `+∞+0i`
• If `z` is `-∞+∞i`, the result is `+∞+3πi/4`
• If `z` is `+∞+∞i`, the result is `+∞+πi/4`
• If `z` is `±∞+NaNi`, the result is `+∞+NaNi`
• 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 `+∞+NaNi`
• If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

The natural logarithm of a complex number z with polar coordinate components (r,θ) equals ln r + i(θ+2nπ), with the principal value ln r + iθ

### Example

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

int main(void)
{
double complex z = clog(I); // r = 1, θ = pi/2
printf("2*log(i) = %.1f%+fi\n", creal(2*z), cimag(2*z));

double complex z2 = clog(sqrt(2)/2 + sqrt(2)/2*I); // r = 1, θ = pi/4
printf("4*log(sqrt(2)/2+sqrt(2)i/2) = %.1f%+fi\n", creal(4*z2), cimag(4*z2));

double complex z3 = clog(-1); // r = 1, θ = pi
printf("log(-1+0i) = %.1f%+fi\n", creal(z3), cimag(z3));

double complex z4 = clog(conj(-1)); // or clog(CMPLX(-1, -0.0)) in C11
printf("log(-1-0i) (the other side of the cut) = %.1f%+fi\n", creal(z4), cimag(z4));
}```

Output:

```2*log(i) = 0.0+3.141593i
4*log(sqrt(2)/2+sqrt(2)i/2) = 0.0+3.141593i
log(-1+0i) = 0.0+3.141593i
log(-1-0i) (the other side of the cut) = 0.0-3.141593i```
• C11 standard (ISO/IEC 9899:2011):
• 7.3.7.2 The clog functions (p: 195)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• G.6.3.2 The clog functions (p: 543-544)
• G.7 Type-generic math <tgmath.h> (p: 545)
• C99 standard (ISO/IEC 9899:1999):
• 7.3.7.2 The clog functions (p: 176-177)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• G.6.3.2 The clog functions (p: 478-479)
• G.7 Type-generic math <tgmath.h> (p: 480)