|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
partial fraction series for digamma function
|
(Theorem)
|
|
|
Theorem 1 $$ \psi (z) = - \gamma -\frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{k} - \frac{1}{z + k} \right)=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{z+k}\right $$
Proof: Start with $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^{z/k}, $$ so $$ \ln\Gamma(z)=-\gamma z - \ln z +\sum_{k=1}^{\infty}\left(-\ln\left(1+\frac{z}{k}\right)+\frac{z}{k}\right $$ and thus, taking derivatives, $$ \psi(z)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(-\frac{1/k}{1+\frac{z}{k}}+\frac{1}{k}\right) =-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\left(\frac{1}{k}-\frac{1}{z+k}\right $$ The second formula follows after rearranging terms (the
rearrangement is legal since we are simply exchanging adjacent terms, so partial sums remain the same).
|
"partial fraction series for digamma function" is owned by rm50.
|
|
(view preamble | get metadata)
Cross-references: partial sums, adjacent, terms, formula, derivatives, proof
This is version 3 of partial fraction series for digamma function, born on 2006-11-11, modified 2007-03-13.
Object id is 8540, canonical name is PartialFractionSeriesForDigammaFunction.
Accessed 913 times total.
Classification:
| AMS MSC: | 33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions) | | | 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
