Dilogarithm of Square/Proof 1

Theorem

$\map {\Li_2} z + \map {\Li_2} {-z} = \dfrac 1 2 \map {\Li_2} {z^2}$

Proof

\(\ds \map {\Li_2} z + \map {\Li_2} {-z}\)

\(=\)

\(\ds -\paren {\int_0^z \frac {\map \ln {1 - t} } t \rd t + \int_0^z \frac {\map \ln {1 + t} } t \rd t}\)

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

$\blacksquare$