How to Find Non-Analytic Terms in an Expansion?

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SUMMARY

This discussion focuses on identifying non-analytic terms in mathematical expansions, particularly through the use of Laurent series and alternative expansion methods. It emphasizes that while analytic terms can be derived from Taylor expansion coefficients, non-analytic terms require a more systematic approach. The conversation highlights the importance of recognizing cases where logarithmic terms precede Taylor series, particularly noting that ln(x) cannot be expanded around x=0. The Frobenius series is mentioned as a valuable technique for solving differential equations via series substitution.

PREREQUISITES
  • Understanding of Taylor series and their coefficients
  • Familiarity with Laurent series and their applications
  • Knowledge of logarithmic functions and their properties
  • Basic concepts of differential equations and series substitution
NEXT STEPS
  • Research the properties and applications of Laurent series
  • Study the Frobenius series method for solving differential equations
  • Explore the limitations of Taylor series, particularly with logarithmic functions
  • Investigate alternative expansion methods for non-analytic terms
USEFUL FOR

Mathematicians, physicists, and students studying advanced calculus or differential equations who need to understand the distinction between analytic and non-analytic terms in series expansions.

jdstokes
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Suppose I have some expression. The analytic terms in its expansion can always be found from the coefficients of the Taylor expansion. What about the non-analytic terms? Is there some systematic way to obtain them?
 
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Analytic in what sense? How systematic you can be depends on how general and nonanalystic the terms are. There are Laurent series so mae you can us that. It may help to use a different type of expansion or other method on the other terms.
 
There are many cases where a series begins with a logarithm term and then proceeds with the ordinary Taylor series. ln(x) cannot be expanded about x=0, so this is necessary.

This is used to solve differential equations by series substitution. Look up the Frobenius series.
 

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