- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's the fourth problem of the week. Hopefully we'll see some people contribute solutions this time around...last week was a little disappointing in that regard. :-/
This week's problem was again proposed by yours truly.
-----
Problem: Let $g$ be the entire function given by
\[g(z) = e^{z\gamma}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k},\]
where $\gamma=\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right)$ is Euler's constant. Prove the recurrence formula
\[(z+1)g(z+1)=g(z).\]
-----
Remember to read the POTW submission guidlines to find out how to submit your answers!
This week's problem was again proposed by yours truly.
-----
Problem: Let $g$ be the entire function given by
\[g(z) = e^{z\gamma}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)e^{-z/k},\]
where $\gamma=\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right)$ is Euler's constant. Prove the recurrence formula
\[(z+1)g(z+1)=g(z).\]
-----
Remember to read the POTW submission guidlines to find out how to submit your answers!