- #1

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- 9

One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with the argument and powers of it.

Normally, the Taylor series expansion for

*f(x)*is:

[tex]f(x) = \sum\limits_{n = 0}^\infty {\frac{{{f^{(n)}}({x_0})}}{{n!}} \cdot {{(x - {x_0})}^n}} [/tex]

But how does this change for a composite function? How do you expand [itex]f(g(x))[/itex]? Do you just replace

*x*in the sum with

*g(x)*and

*x*

_{0}with

*g(x*? Or do you differentiat

_{0})*f(g(x))*with respect to

*x*in each term before setting

*x*to

*x*

_{0}?

But I suspect the even harder question is how do you take the derivatives of the Dirac delta function for each term in the sum. And if you have to evaluate it at

*x*

_{0}, won't that make each term infinity? Is there an easy expression for [itex]{\delta ^{(n)}}({x_0})[/itex]? Or better yet, how do we evaluate [itex]{\delta ^{(n)}}(g(x))[/itex]? Thank you.