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January 26th, 2016, 11:50 AM   #1
Joined: Jun 2009

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Approximation near the singularity

Hi, I would like to somehow "approximate" function f(x) near its singularity p ($\displaystyle f(p)=\infty$) by a sum of powers of x.
To be more precise, let f be function from R to R (real numbers - but if it would be more convenient, let's use complex numbers instead), let f had as "neat" properties as you wish (eg. let f have one-sided derivatives of any order) in the neighbourhood its singularity p (where $\displaystyle f(p)=\infty$). An example of such a function is $\displaystyle f(x):=1/x$ and p:=0 and we would like to approximate f(x) "near" x=0 (ie. where x>0 and x is "very small").
We would like to approximate f somehow by (from now we will suppose wlog that p=0) "sum" (*) "$\displaystyle \sum_{i=0}^{\infty}(a_i.x^i)$" - but in fact this is not (convergent) sum for any x.
The question is how to precisely define, that the sum (*) approximates f (near p) well and how to find the coefficients a_i. Maybe that some methods of asymptotic analysis (eg asymptotic series) can be used but I would also like to see explicit construction (or general method) of the coefficients $\displaystyle a_i$ (say for functions 1/x and ln(x) - near 0).
Thank you for any help.
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January 26th, 2016, 12:00 PM   #2
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Since the $i=0$ term is the dominant one for small $x$, your $a_i$ clearly do not have well defined values.

What we tend to do with such functions is to determine the Laurent series, which includes negative powers of $(x-p)$. Obviously, this doesn't do much for $f(x)=/frac1x$, but it's useful for Complex Analysis.
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