# Where is the error in the following line (complex logarithm)?

Let's take the principal branch $$L(Re^{i\varphi})=\ln(R)+i\varphi$$ of the logarithm. I got myself confused now over this: Where is the error in the line $$i\pi=L(e^{i\pi})=L(e^{i\pi}\cdot 1)=L(e^{i\pi}\cdot e^{2i\pi})=L(e^{3i\pi})=3i\pi$$

Also: the principal branch isn't even defined for $$e^{i\pi}$$ but still we can compute it there. What's going on exactly?

• If you look at the principal branch, then $L_{principal}(e^{3i\pi})=i\pi$. You are not allowed to simply "cancell" $L$ with $e$. :) – C. Brendel May 8 at 10:09

## 1 Answer

In the last equality: $$L(e^{3i\pi})=i\pi$$, not $$3i\pi$$. In the definition of principal branch of the algorithm, we assume that $$\varphi\in(-\pi,\pi)$$ when we del with the expression $$L(Re^{i\varphi})=\ln R+i\varphi$$. Note that otherwise the definition would be ambiguous.