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Math Help - Direct Analytical Continuations

  1. #1
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    Direct Analytical Continuations

    Hi any help with this would be greatly appreciated:

    Direct Analytical Continuations-q3a.jpg

    Many thanks and "Mathshelpforum" rocks
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  2. #2
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    Quote Originally Posted by moolimanj View Post
    Hi any help with this would be greatly appreciated:

    Click image for larger version. 

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    Many thanks and "Mathshelpforum" rocks
    <br />
f(z)=\sum_{n=0}^{\infty}(2+z)^n, \ \ (|z+2|<1)<br />

    <br />
g(z)=-2 \sum_{n=1}^{\infty}\left( \frac{1}{3+2x}\right)^n, \ \ (|z+3/2|>1/2)<br />

    The series defining function f(z) is a geometric series which sums to:

    <br />
f(z)=\frac{-1}{1+z}, \ \ (|z+2|<1)<br />

    Similarly the series defining g(z) may also be summed with the
    aid of the geometric series formula to:


    <br />
g(z)=2-2 \sum_{n=0}^{\infty}\left( \frac{1}{3+2x}\right)^n<br />
=2-\frac{2}{1-\frac{1}{3+2z}}=\frac{-1}{1+z} , \ \ (|z+3/2|>1/2)<br />

    Thus f(z) and g(z) are identical on the overlap of their regions of convergence, and so are analytic continuations of one another.

    RonL
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