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 January 26th, 2016, 11:50 AM #1 Member   Joined: Jun 2009 Posts: 83 Thanks: 1 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. January 26th, 2016, 12:00 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 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. Tags approximation, asymptotics, singularity Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post aaron-math Complex Analysis 3 October 16th, 2013 04:51 PM mathbalarka Calculus 5 April 14th, 2012 01:48 PM Wissam Number Theory 16 March 13th, 2011 04:41 PM bull-roarer Algebra 4 May 21st, 2009 11:16 AM goldgrill Complex Analysis 0 November 18th, 2007 11:34 PM

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