Engineering Mathematics: Series convolution

Sunday, April 3, 2011

Series convolution

SC.ID.1: linear series convolution

$\left ( \sum _{n=0}^{\infty}a_{n}x^{n} \right )\left ( \sum _{n=0}^{\infty}b_{n}x^{n} \right )=\sum _{n=0}^{\infty}c_{n}x^{n}$

where

$c_{n}=\sum _{k=0}^{n}a_{k}b_{n-k}$

SC.ID.2: exponential series convolution

$\left ( \sum _{n=0}^{\infty}\frac{a_{n}}{n!}x^{n} \right )\left ( \sum _{n=0}^{\infty}\frac{b_{n}}{n!}x^{n} \right )=\left ( \sum _{n=0}^{\infty}\frac{c_{n}}{n!}x^{n} \right )$

where

$\frac{c_{n}}{n!}=\sum _{k=0}^{n}\frac{a_{k}}{k!}\frac{b_{n-k}}{(n-k)!}$

therefore

$c_{n}=\sum _{k=0}^{n}\frac{n!}{k!(n-k)!}a_{k}b_{n-k}=\sum _{k=0}^{n}\binom{n}{k}a_{k}b_{n-k}$

Engineering Mathematics

Sunday, April 3, 2011

Series convolution

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