example of Cauchy multiplication rule

Let us form the Taylor expansion of  $e^x\sin y$  starting from the known Taylor expansions

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\mathrm{\dots },$$

$$\sin y=y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}+-\mathrm{\dots }$$

and multiplying these series with Cauchy multiplication rule. As power series, both series are absolutely convergent for all real (and complex) values of $x$ and $y$. The rule gives immediately the series

$y+(-{\displaystyle \frac{y^3}{3!}}+xy)+({\displaystyle \frac{y^5}{5!}}-{\displaystyle \frac{x{y}^3}{3!}}+{\displaystyle \frac{x^{2}y}{2!}})+(-{\displaystyle \frac{y^7}{7!}}+{\displaystyle \frac{x{y}^5}{5!}}-{\displaystyle \frac{x^2{y}^3}{2!3!}}+{\displaystyle \frac{x^{3}y}{3!}})+({\displaystyle \frac{y^9}{9!}}-{\displaystyle \frac{x{y}^7}{7!}}+{\displaystyle \frac{x^2{y}^5}{2!5!}}-{\displaystyle \frac{x^3{y}^3}{3!3!}}+{\displaystyle \frac{x^{4}y}{4!}})+\mathrm{\dots }$ (1)

The parenthesis expressions here seem a bit irregular, but we can regroup and rearrange the terms in new parentheses:

$e^x\sin y=y+{\displaystyle \frac{xy}{1!1!}}+\left({\displaystyle \frac{x^{2}y}{2!1!}}-{\displaystyle \frac{y^3}{3!}}\right)+\left({\displaystyle \frac{x^{3}y}{3!1!}}-{\displaystyle \frac{x{y}^3}{1!3!}}\right)+\left({\displaystyle \frac{x^{4}y}{4!1!}}-{\displaystyle \frac{x^2{y}^3}{2!3!}}+{\displaystyle \frac{y^5}{5!}}\right)+\mathrm{\dots }$ (2)

It’s clear that the last series precisely the same terms as the preceding one. The regrouping and the rearranging of the terms is allowable, since also (1) is converges absolutely. In fact, if one would multiply the series of $e^x$ with the series $y+\frac{y^3}{3!}+\frac{y^5}{5!}+\mathrm{\dots }$ of $\sinh y$ (which converges absolutely $\forall x\in ℝ$), one would get the series like (1) but all signs “+”; by the Cauchy multiplication rule this series converges especially for each positive $x$ and $y$, in which case it is a series with positive terms; hence (1) is absolutely convergent.

The form (2) can be obtained directly from the Taylor series formula (http://planetmath.org/TaylorSeries).

Title	example of Cauchy multiplication rule
Canonical name	ExampleOfCauchyMultiplicationRule
Date of creation	2013-03-22 17:29:19
Last modified on	2013-03-22 17:29:19
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Example
Classification	msc 41A58
Classification	msc 40-00
Classification	msc 30B10
Classification	msc 26A24
Synonym	Taylor series gotten by multiplication
Related topic	HyperbolicFunctions