# Dilogarithm of Square

$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$
where $\Li_2$ denotes the dilogarithm function.
 $\ds \map {\Li_2} z + \map {\Li_2} {-z}$ $=$ $\ds -\paren {\int_0^z \frac {\map \ln {1 - t} } t \rd t + \int_0^z \frac {\map \ln {1 + t} } t \rd t}$ Definition of Dilogarithm Function $\ds$ $=$ $\ds -\int_0^z \frac {\map \ln {\paren {1 - t} \paren {1 + t} } } t \rd t$ Linear Combination of Definite Integrals, Sum of Logarithms $\ds$ $=$ $\ds -\int_0^z \frac {\map \ln {1 - t^2} } t \rd t$ Difference of Two Squares $\ds$ $=$ $\ds -\int_0^{z^2} \frac {\map \ln {1 - t^2} } t \frac {\map \d {t^2} } {2 t}$ substituting $t \to t^2$ $\ds$ $=$ $\ds -\frac 1 2 \int_0^{z^2} \frac {\map \ln {1 - t^2} } {t^2} \map \rd {t^2}$ $\ds$ $=$ $\ds \frac 1 2 \map {\Li_2} {z^2}$ Definition of Dilogarithm Function
$\blacksquare$