Dilogarithm of Zero

Theorem

$\map {\Li_2} 0 = 0$

where:

$\map {\Li_2} x$ is the dilogarithm function of $x$

Proof

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

\(=\)

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

Power Series Expansion for Spence's Function

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {0^n} {n^2}\)

$z := 0$

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$