# std::log(std::complex)

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< cpp‎ | numeric‎ | 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)

### 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 | computes natural (base e) logarithm (to base e) (ln(x)) (function)
log(std::valarray) | applies the function std::log to each element of valarray (function template)
C documentation for clog

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