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

Notes

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

Run this code

#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

References

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)

Language
Headers
Type support
Program utilities
Variadic function support
Error handling
Dynamic memory management
Date and time utilities
Strings library
Algorithms
Numerics
Input/output support
Localization support
Atomic operations (C11)
Thread support (C11)
Technical Specifications

Common mathematical functions
Floating-point environment
Complex number arithmetics
Random number generation
Type-generic math

cexpcexpfcexpl (C99)(C99)(C99)	computes the complex base-e exponential (function)
loglogflogl (C99)(C99)	computes natural (base-e) logarithm ( $ln(x)$ ) (function)