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.Q:
$e^{ -\pi i\int \frac{1}{x}\,dx} = \frac{1}{x}+C$
I was working on a problem. I am aware that there are simpler methods, but it is the final solution to a problem I am using.
$$e^{ -\pi i\int \frac{1}{x}\,dx}=
\begin{cases}
\phantom{ – }\frac{1}{x}+C
&\quad,\text{ if }x
eq0,
\\
-\frac{\pi}{2}
&\quad,\text{ if }x=0,
\end{cases}$$
I found this solution below.
Here, he is utilizing the fact that $\sin z=z\cos z$ and $\cos z=1-2\sin^2\frac{z}{2}$. I get the first part, but I don’t get the second part. Can somebody please explain why that is?
A:
The value of $\int_{0}^{1}\frac{1}{x}\,dx$ is $\pi/2$. Therefore, $-\frac{\pi}{2}$ is the correct result.
A:
I suspect your professor had a typo. Perhaps he meant
$$e^{ -\pi i\left(-\pi^2\right)}=\color{red}{\frac{\pi}{x}+C}$$
which integrates to
$$-\frac{\pi^2}{2}$$
where $\frac{\pi^2}{2} https://ourlittlelab.com/wp-content/uploads/2022/06/ellydelc.pdf
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