Series expansion of an integral

[praharmitra]

Given a function g(t), define the function f(r) as follows

$$f(r) = \int_r^\infty g(t) dt$$

I want to find the series expansion of f(r) around the point r = 0, without actually doing the integral. Is this possible?

Basically, can i use any particular series expansion of g to find the series expansion of f?

 

[tiny-tim]

hi praharmitra!

you want to find every dⁿf(r)/drⁿ ?

hint: what is df(r)/dr ? ​

 

[praharmitra]

hey tinytim!

I'm afraid we cannot use the usual expression for taylor expansions. You see, the function g(t) that I have does not converge on r=0. I believe there is a 1/r^2 divergence at r=0. So, the first term in the taylor series expansion, which is f(0) is not defined, so we cannot use what u have said.

I was thinking, we could use laurent expansion, but am unsure how to use it.

 

[micromass]

Can you give us what the function g is? Or is this function to complicated?

 

[praharmitra]

Can you give us what the function g is? Or is this function to complicated?

I don't think I can. Mathematica gave me a 10 line expression for the function. Its not able to integrate it either. However, it can series expand it about the usual values (0, infinity, 1).

So, i was hoping we could figure out the series expansion of f using the series expansion of g.

I'll tell you exactly what I need:

I believe f will diverge about r = 0 as 1/r^2 (from experience on previous similar calculations all of which i could integrate). All I want actually is the coefficient of the 1/r^2 in the expansion and the constant term. I don't want the full expansion really.