Complex logarithm

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:

• A complex logarithm of a nonzero complex number ${\displaystyle z}$, defined to be any complex number ${\displaystyle w}$ for which ${\displaystyle e^{w}=z}$.^[1][2] Such a number ${\displaystyle w}$ is denoted by ${\displaystyle \log z}$.^[1] If ${\displaystyle z}$ is given in polar form as ${\displaystyle z=re^{i\theta }}$, where ${\displaystyle r}$ and ${\displaystyle \theta }$ are real numbers with ${\displaystyle r>0}$, then ${\displaystyle \ln r+i\theta }$ is one logarithm of ${\displaystyle z}$, and all the complex logarithms of ${\displaystyle z}$ are exactly the numbers of the form ${\displaystyle \ln r+i\left(\theta +2\pi k\right)}$ for integers ${\displaystyle k}$.^[1][2] These logarithms are equally spaced along a vertical line in the complex plane.
• A complex-valued function ${\displaystyle \log \colon U\to \mathbb {C} }$, defined on some subset ${\displaystyle U}$ of the set ${\displaystyle \mathbb {C} ^{*}}$ of nonzero complex numbers, satisfying ${\displaystyle e^{\log z}=z}$ for all ${\displaystyle z}$ in ${\displaystyle U}$. Such complex logarithm functions are analogous to the real logarithm function ${\displaystyle \ln \colon \mathbb {R} _{>0}\to \mathbb {R} }$, which is the inverse of the real exponential function and hence satisfies e^{ln x} = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of ${\displaystyle 1/z}$, or by the process of analytic continuation.

There is no continuous complex logarithm function defined on all of ${\displaystyle \mathbb {C} ^{*}}$. Ways of dealing with this include branches, the associated Riemann surface, and partial inverses of the complex exponential function. The principal value defines a particular complex logarithm function ${\displaystyle \operatorname {Log} \colon \mathbb {C} ^{*}\to \mathbb {C} }$ that is continuous except along the negative real axis; on the complex plane with the negative real numbers and 0 removed, it is the analytic continuation of the (real) natural logarithm.