Power Series Expansion for Spence's Function
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Theorem
Spence's function has a power series expansion:
- $\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$
for $z \in \C$.
This converges for $\size z \le 1$.
Proof
\(\ds \map {\Li_2} z\) | \(=\) | \(\ds -\int_0^z \frac {\map \ln {1 - t} } t \rd t\) | Definition of Spence's Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_0^z \frac 1 t \sum_{n \mathop = 1}^\infty \paren {-\frac {t^n} n} \rd t\) | Power Series Expansion for Logarithm of 1 + x: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \int_0^z \frac {t^{n - 1} } n \rd t\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}\) | Primitive of Power |
$\blacksquare$