std::log(std::complex)

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.

If z is a negative real number, std::imag(std::log(x)) equals pi.

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 $ln r + iθ$

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)

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