Definite Integral over Unit Square of Logarithm of x minus Logarithm of y over x minus y
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Theorem
- $\ds \int_0^1 \int_0^1 \frac {\ln x - \ln y} {x - y} \rd x \rd y = 2 \map \zeta 2$
Proof
| \(\ds \int_0^1 \int_0^1 \dfrac {\ln x - \ln y} {x - y} \rd x \rd y\) | \(=\) | \(\ds \int_0^1 \int_0^1 \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y\) | Difference of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^1 \int_0^x \dfrac {\map \ln {\dfrac x y} } {x - y} \rd x \rd y\) | symmetry about the line $y = x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^1 \int_0^1 \dfrac {\map \ln {\dfrac x {x t} } } {x - x t} \rd x \paren {x \rd t}\) | $y \to x t$ and $\rd y \to x \rd t$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^1 \int_0^1 \dfrac {\map \ln {\dfrac 1 t } } {\paren {1 - t} } \rd x \rd t\) | canceling the $x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^1 \dfrac {\map \ln {\dfrac 1 t } } {\paren {1 - t} } \rd t \bigintlimits x 0 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \int_0^1 \dfrac {\map \ln t } {\paren {1 - t} } \rd t\) | Logarithm of Reciprocal | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \int_1^0 \dfrac {\map \ln {1 - x} } x \paren {-\rd x}\) | $\paren {1 - t} \to x$ and $-\rd t \to \rd x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \int_0^1 \dfrac {\map \ln {1 - x} } x \rd x\) | reversing limits of integration | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map {\Li_2} 1\) | Definition of Spence's Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \sum_{n \mathop = 1}^\infty \frac {1^n} {n^2}\) | Power Series Expansion for Spence's Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map \zeta 2\) | Definition of Riemann Zeta Function |
$\blacksquare$