:PROPERTIES: :ID: fa9fa179-f4ea-4af2-914b-658fcf638969 :mtime: 20231024012008 :ctime: 20231024011949 :END: #+title: complex logarithm #+filetags: :public:project: * Definition The complex logarithm refers to the taking the logarithm of a [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex number]]. We want to find all $q \in \mathbb{C}$ such that \[e^{q} = z\] for a given $z \in \mathbb{C}$. \[\log(z) = \log(|z|) + i(\theta = 2\pi n)\] * Non-uniqueness Note that the logarithm of a complex number is NOT unique. The real part is unique, but you may always add $2\pi i$ to the imaginary part. * Equations about complex logarithm \[Re(z) = \log(|z|)\] \[Im(z) = arg(z)\] * Example Problems ** Example 1: $2i$ *** Question Find $\log(2i)$ *** Solution \[\log(2i) = \log(2) + i\pi(2n+\frac{1}{2})\] ** Example 2: $2^i$ *** Question What is $2^i$ *** Solution \[\log(2^{i}) = i \log(2)\] \[2^{i} = \cos(\log(2)) + i\sin(\log(2))\] ** Example 3: $i^i$ *** Question What is $i^i$ *** Solution \[\log(i^{i}) = i\log(i)\] ** Example 4: $r\exp(i\theta)^{x+iy}$ *** Question What is $r\exp(i\theta)^{x+iy}$ *** Solution \begin{align} &r \exp(i\theta)^{x + iy} \\ &= \end{align}