NAME¶

std::log(std::complex) - std::log(std::complex)

Synopsis¶

Defined in header <complex>
template< class T >
complex<T> log( const complex<T>& z );

Computes complex natural (base e) logarithm of a complex value z with a branch cut
along the negative real axis.

Parameters¶

z - complex value

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
* std::log(std::conj(z)) == std::conj(std::log(z))
* If z is (-0,+0), the result is (-∞,π) and FE_DIVBYZERO is raised
* If z is (+0,+0), the result is (-∞,+0) and FE_DIVBYZERO is raised
* If z is (x,+∞) (for any finite x), the result is (+∞,π/2)
* If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may
be raised
* If z is (-∞,y) (for any finite positive y), the result is (+∞,π)
* If z is (+∞,y) (for any finite positive y), the result is (+∞,+0)
* If z is (-∞,+∞), the result is (+∞,3π/4)
* If z is (+∞,+∞), the result is (+∞,π/4)
* If z is (±∞,NaN), the result is (+∞,NaN)
* If z is (NaN,y) (for any finite y), the result is (NaN,NaN) and FE_INVALID may
be raised
* If z is (NaN,+∞), the result is (+∞,NaN)
* If z is (NaN,NaN), the result is (NaN,NaN)

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

The semantics of this function are intended to be consistent with the C function
clog.

Defect reports

The following behavior-changing defect reports were applied retroactively to
previously published C++ standards.

DR Applied to Behavior as published Correct behavior
LWG 2597 C++98 specification mishandles signed zero erroneous requirement
imaginary parts removed

Example¶

// Run this code

#include <iostream>
#include <cmath>
#include <complex>

int main()
{
std::complex<double> z{0, 1}; // r = 1, θ = pi/2
std::cout << "2*log" << z << " = " << 2.*std::log(z) << '\n';

std::complex<double> z2{sqrt(2)/2, sqrt(2)/2}; // r = 1, θ = pi/4
std::cout << "4*log" << z2 << " = " << 4.*std::log(z2) << '\n';

std::complex<double> z3{-1, 0}; // r = 1, θ = pi
std::cout << "log" << z3 << " = " << std::log(z3) << '\n';
std::complex<double> z4{-1, -0.0}; // the other side of the cut
std::cout << "log" << z4 << " (the other side of the cut) = " << std::log(z4) << '\n';
}

Output:¶

2*log(0,1) = (0,3.14159)
4*log(0.707107,0.707107) = (0,3.14159)
log(-1,0) = (0,3.14159)
log(-1,-0) (the other side of the cut) = (0,-3.14159)

See also¶

complex common logarithm with the branch cuts along the negative
log10(std::complex) real axis
(function template)
exp(std::complex) complex base e exponential
(function template)
log
logf computes natural (base e) logarithm (\({\small \ln{x} }\)ln(x))
logl (function)
(C++11)
(C++11)
log(std::valarray) applies the function std::log to each element of valarray
(function template)