Taylor Series of Logarithm of Gamma Function

Theorem
Let $\gamma$ denote the Euler-Mascheroni constant.

Let $\map \zeta s$ denote the Riemann zeta function.

Let $\map \Gamma z$ denote the gamma function.

Let $\Log$ denote the natural logarithm.

Then $\map \Log {\map \Gamma z}$ has the power series expansion:

which is valid for all $z \in \C$ such that $\cmod {z - 1} < 1$.

Proof
From Gamma Difference Equation:
 * $\map \Gamma {z + 1} = z \map \Gamma z$

Hence:

Successive use of this identity gives us

and thus from the Sum Rule for Derivatives:

From Stirling's Formula for Gamma Function:

which by definition of the Euler-Mascheroni constant:


 * $\displaystyle \frac {\d} {\d z} \map \Log {\map \Gamma 1} = \lim_{M \to \infty} \frac {\d} {\d z} \map \Log {\map \Gamma {1 + M} } - \sum_{k \mathop = 0}^{M - 1} \frac 1 {1 + k} = -\gamma$

Also:
 * $\dfrac {\d^{1 + k} } {\d z^{1 + k}} \map \Log {\map \Gamma {z + 1} } = \map \OO {\frac 1 z}$

shows that:
 * $\displaystyle \lim_{M \to \infty} \frac {\d^{1 + k} } {\d z^{1 + k} } \map \Log {\map \Gamma {M + 1} } = 0$

thus for $n > 1$:

Thus by definition of Taylor series:

From Zeroes of Gamma Function, we see that $\map \Gamma z$ is non-zero everywhere.

Thus $\map \Log {\map \Gamma z}$ has poles only where $\Gamma$ does, that is, the negative integers.

Since the radius of convergence of a power series is equal to the distance of its center to the closest point where the function is not analytic:


 * The radius of convergence of $\map \Log {\map \Gamma z}$ is $\cmod {1 - 0} = 1$.