Power Series Expansion for Logarithm of Gamma Function
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Theorem
The logarithm of the Gamma function has the power series expansion:
\(\ds \map \ln {\map \Gamma {x + 1} }\) | \(=\) | \(\ds -\gamma x + \sum_{k \mathop = 2}^\infty \dfrac {\map \zeta k \paren {-x}^k} k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma x + \dfrac {\map \zeta 2 x^2} 2 - \dfrac {\map \zeta 3 x^3} 3 + \dfrac {\map \zeta 4 x^4} 4 - \dfrac {\map \zeta 5 x^5} 5 + \cdots\) |
valid for all $x \in \C$ such that $\size x < 1$ and $x = 1$.
Proof
\(\ds \gamma + \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }\) | \(=\) | \(\ds \map H x\) | Extension of Harmonic Number to Non-Integer Argument | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }\) | \(=\) | \(\ds -\gamma + \map H x\) | subtracting $\gamma$ from both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int_0^x \frac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} } \rd x\) | \(=\) | \(\ds \int_0^x \paren {-\gamma + \map H x} \rd x\) | integrating both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {\map \Gamma {x + 1} }\) | \(=\) | \(\ds \int_0^x \paren {-\gamma + \map H x} \rd x\) | Definition of Digamma Function and the Fundamental Theorem of Calculus | ||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^x \paren {-\gamma + \sum_{k \mathop = 2}^\infty \paren {-1}^k \map \zeta k x^{k - 1} } \rd x\) | Power Series Expansion for Harmonic Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {-\gamma x + \sum_{k \mathop = 2}^\infty \dfrac {\map \zeta k \paren {-x}^k} k} 0 x\) | Primitive of Power and Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma x + \sum_{k \mathop = 2}^\infty \dfrac {\map \zeta k \paren {-x}^k} k\) |
When $x = 1$, we have $\sequence {a_k = \dfrac {\map \zeta k} k}_{k \mathop \ge 0}$ a decreasing sequence of positive terms in $\R$ which converges with a limit of zero.
By the Alternating Series Test, this alternating series converges at $x = 1$.
$\blacksquare$
Sources
- 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $7$. Sums of Powers, Bernoulli Numbers, and the Gamma Function