Power Series Expansion for Spence's Function

Theorem

Spence's function has a power series expansion:

$\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$

This converges for $\size z \le 1$.

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Proof

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

\(=\)

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

Definition of Spence's Function

\(\ds \)

\(=\)

\(\ds -\int_0^z \frac 1 t \sum_{n \mathop = 1}^\infty \paren {-\frac {t^n} n} \rd t\)

Corollary to Power Series Expansion for $\map \ln {1 + x}$

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \int_0^z \frac {t^{n - 1} } n \rd t\)

Fubini's Theorem

\(\ds \)

\(=\)

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

Primitive of Power

$\blacksquare$