- #1
aheight
- 321
- 109
- Homework Statement
- I am having problems showing an integral converges to -Pi^2/6
- Relevant Equations
- Li_2(1)
I am working with the Dilogarithm function and am having problems showing the following and was wondering if someone could help:
$$
\int_0^1\int_0^y\left(\frac{1}{x-1}\right)\left(\frac{1}{y}\right)dxdy=-\frac{\pi^2}{6}
$$
This is what I have so far:
Iterating the first level:
$$
\begin{align*}
&=\int_0^1\frac{1}{y}\biggr(\text{Log}(x-1)\biggr|_0^y\biggr)dy\\
&=\int_0^1\frac{1}{y}\biggr(\frac{\text{Log}(y-1)-\pi i}{y}\biggr)dy
\end{align*}
$$
which is now improper so I can write:
$$
\begin{align*}
&=\lim_{\epsilon\to 0^+}\int_{\epsilon}^{1-\epsilon}\frac{\text{Log}(y-1)-\pi i}{y}dy\\
&=\lim_{\epsilon\to 0^+}\biggr\{\int_{\epsilon}^{1-\epsilon}\frac{\text{Log}(y-1)}{y}dy+\pi i\text{Log}(\epsilon)\biggr\}
\end{align*}
$$
and an this point I'm stuck. I realize the integral is equal to the special function ##-\text{Li}_2(1)## but I'd like to prove it by evaluating the limit above unless there is a more standard approach of doing so. Perhaps a more straight-forward question would be "How does one show ##\text{Li}_2(1)=-\frac{\pi^2}{6}##?"
MENTOR NOTE: see post #6 for a correction to this post.
$$
\int_0^1\int_0^y\left(\frac{1}{x-1}\right)\left(\frac{1}{y}\right)dxdy=-\frac{\pi^2}{6}
$$
This is what I have so far:
Iterating the first level:
$$
\begin{align*}
&=\int_0^1\frac{1}{y}\biggr(\text{Log}(x-1)\biggr|_0^y\biggr)dy\\
&=\int_0^1\frac{1}{y}\biggr(\frac{\text{Log}(y-1)-\pi i}{y}\biggr)dy
\end{align*}
$$
which is now improper so I can write:
$$
\begin{align*}
&=\lim_{\epsilon\to 0^+}\int_{\epsilon}^{1-\epsilon}\frac{\text{Log}(y-1)-\pi i}{y}dy\\
&=\lim_{\epsilon\to 0^+}\biggr\{\int_{\epsilon}^{1-\epsilon}\frac{\text{Log}(y-1)}{y}dy+\pi i\text{Log}(\epsilon)\biggr\}
\end{align*}
$$
and an this point I'm stuck. I realize the integral is equal to the special function ##-\text{Li}_2(1)## but I'd like to prove it by evaluating the limit above unless there is a more standard approach of doing so. Perhaps a more straight-forward question would be "How does one show ##\text{Li}_2(1)=-\frac{\pi^2}{6}##?"
MENTOR NOTE: see post #6 for a correction to this post.
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