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

hey bestie!! welcome to std::log(std::complex)(3) from the C++ Standard Libary manual — this is the C++ standard library page and it's honestly giving main-character math energy 💅✨

🎀 NAME 🎀

std::log(std::complex) — std::log(std::complex)

✨ Synopsis ✨

Defined in header <complex>

template< class T >
std::complex<T> log( const std::complex<T>& z );

Computes the 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.

🦋 Example 🦋

// Run this code

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

int main()
{
    std::complex<double> z {0.0, 1.0}; // r = 1, θ = pi / 2
    std::cout << "2 * log" << z << " = " << 2.0 * std::log(z) << '\n';

    std::complex<double> z2 {sqrt(2.0) / 2, sqrt(2.0) / 2}; // r = 1, θ = pi / 4
    std::cout << "4 * log" << z2 << " = " << 4.0 * std::log(z2) << '\n';

    std::complex<double> z3 {-1.0, 0.0}; // r = 1, θ = pi
    std::cout << "log" << z3 << " = " << std::log(z3) << '\n';
    std::complex<double> z4 {-1.0, -0.0}; // the other side of the cut
    std::cout << "log" << z4 << " (the other side of the cut) = " << std::log(z4) << '\n';
}

Possible 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)

💀 Defect reports 💀

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

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

🌈 See also 🌈

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

---

okay that's the whole page 🫶 — sourced from http://cppreference.com, dated 2024.06.10, and it's the std::log(std::complex)(3) page. periodt. 💖