Dilogarithm of Square

Theorem

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

where $\Li_2$ denotes the dilogarithm function.

Proof 1

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

\(=\)

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

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

$\blacksquare$

Proof 2

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

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2} + \sum_{n \mathop = 1}^\infty \frac {\paren {-z}^n} {n^2}\)

Power Series Expansion for Spence's Function

\(\ds \)

\(=\)

\(\ds \paren {z + \frac {z^2} {2^2} + \frac {z^3} {3^2} + \frac {z^4} {4^2} + \frac {z^5} {5^2} + \frac {z^6} {6^2} + \cdots} + \paren {-z + \frac {z^2} {2^2} - \frac {z^3} {3^2} + \frac {z^4} {4^2} - \frac {z^5} {5^2} + \frac {z^6} {6^2} + \cdots}\)

\(\ds \)

\(=\)

\(\ds 2 \paren {\frac {z^2} {2^2} + \frac {z^4} {4^2} + \frac {z^6} {6^2} + \cdots}\)

odd terms cancel, even terms double

\(\ds \)

\(=\)

\(\ds \frac 2 {2^2} \paren {\frac {z^2} {1^2} + \frac {z^4} {2^2} + \frac {z^6} {3^2} + \cdots}\)

factoring out $2^2$

\(\ds \)

\(=\)

\(\ds \frac 1 2 \sum_{n \mathop = 1}^\infty \frac {\paren {z^2}^n} {n^2}\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \map {\Li_2} {z^2}\)

Definition of Dilogarithm Function

$\blacksquare$