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benorin
#50
Jan23-06, 04:56 AM
HW Helper
P: 1,025
A problem: From the following

i. [tex]\lim_{q\rightarrow\infty}\prod_{n=p}^{pq} \left(1+\frac{x}{n}\right)=p^{x},[/tex]

and

ii.[tex]\lim_{n\rightarrow\infty}\left(1-\frac{t}{n}\right)^{n}=e^{-t},[/tex]

reason that

[tex]\int_{t=0}^{\infty}e^{-t}t^{x}dt=\lim_{n\rightarrow\infty}\frac{1\cdot 2\cdots n}{(1+x)(2+x)\cdots (n+x)}n^{x},[/tex]

where x is complex and not a negative integer.

I quoted this exercise from Introduction to the Theory of Analytic Functions by Harkness & Morley that I happed to have a single page of printed (pg. 208), (i) is the from a seperate exercise listed immediately prior to the exercise at hand in which (ii) is a given.