std::log(std::complex)(3)

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)