TITLE: Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$
QUESTION [4 upvotes]: Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$
Is it equal to $0$ ? Why ?
Any hint ?

REPLY [0 votes]: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\bbox[5px,#ffd]{}}$

\begin{align}
&\bbox[5px,#ffd]{\int_{0}^{2\pi}
\ln\pars{\verts{\expo{\ic\theta} - 1}\dd\theta}} =
2\,\Re\int_{0}^{2\pi}
\ln\pars{\expo{\ic\theta} - 1}\dd\theta
\\[5mm] = &\
2\,\Re\oint_{\verts{z}\ =\ 1}\,\,
\ln\pars{z - 1}{\dd z \over \ic z} =
2\,\Im\oint_{\verts{z}\ =\ 1}\,\,
{\ln\pars{z - 1} \over z}\,\dd z
\\[5mm] \stackrel{z\ \mapsto\ z - 1}{=}\,\,\,\,\,&
2\,\Im\oint_{\verts{z + 1}\ =\ 1}\,\,
{\ln\pars{z} \over z + 1}\,\dd z
\end{align}
I'll choose the $\ds{\ln}$-branch cut
$$
\ln\pars{z} = \ln\pars{\verts{z}} +
\ic\arg\pars{z}\,,\quad
0 < \arg\pars{z} < 2\pi\,,\quad z \not= 0
$$
Then $\ds{\pars{~\mbox{with an}\ indent\ \mbox{around the origin}~}}$,
\begin{align}
&\bbox[5px,#ffd]{\int_{0}^{2\pi}
\ln\pars{\verts{\expo{\ic\theta} - 1}\dd\theta}}
\\ = &\ \
\overbrace{2\,\Im\braces{2\pi\ic\bracks{\ln\pars{\verts{-1}} + \ic\pi}}}^{\ds{=\ 0}}\ -\
\overbrace{2\,\Im\lim_{\epsilon \to 0^{+}}\,\,
\int_{2\pi}^{0}\bracks{\ln\pars{\epsilon} + \ic\theta}
\epsilon\expo{\ic\theta}\ic\,\dd\theta}^{\ds{=\ 0}}
\\[5mm] = & \ \bbx{0} \\ &
\end{align}
{"set_name": "stack_exchange", "score": 4, "question_id": 679233}