MathDB
analytic function

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February 26, 2011
functiongeometryadvanced fieldsadvanced fields unsolved

Problem Statement

a) prove that the function ζ(s)=∑n=1∞1ns\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s} that is defined on the area Re(s)>1Re(s)>1, is an analytic function.
b) prove that the function ζ(s)−1s−1\zeta(s)-\frac{1}{s-1} can be spanned to an analytic function over C\mathbb C.
c) using the span of part b show that ζ(1−n)=−Bnn\zeta(1-n)=-\frac{B_n}{n} that BnB_n is the nnth bernoli number that is defined by generating function tet−1=∑n=0∞Bntnn!\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}.
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