How to Find Non-Analytic Terms in an Expansion?

In summary, the coefficients of the Taylor expansion can determine the analytic terms in an expression, but for non-analytic terms, a different type of expansion such as Laurent series or the Frobenius series may be necessary. This is often used to solve differential equations through series substitution and can involve logarithmic terms.
  • #1
jdstokes
523
1
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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  • #2
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.
 
  • #3
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.
 

FAQ: How to Find Non-Analytic Terms in an Expansion?

What are non-analytic terms in an expansion?

Non-analytic terms in an expansion refer to terms that cannot be expressed as a finite sum of powers of a variable. They typically contain logarithmic, exponential, or trigonometric functions.

Why are non-analytic terms important in scientific research?

Non-analytic terms often appear in physical and mathematical models, and understanding their behavior is crucial for accurate predictions and interpretations. They can also reveal relationships between different variables and provide insight into underlying mechanisms.

How do non-analytic terms affect the convergence of an expansion?

Non-analytic terms can significantly alter the convergence properties of an expansion, as they may introduce singularities or oscillations that can lead to slower or even non-convergent series. It is important to carefully consider the presence of non-analytic terms when working with expansions.

Can non-analytic terms be approximated by analytic functions?

In some cases, non-analytic terms can be approximated by analytic functions to simplify calculations or improve convergence. However, this approximation may introduce errors, and the accuracy of the results should be carefully evaluated.

How can non-analytic terms be handled in practical applications?

In practical applications, non-analytic terms can be dealt with using various techniques such as truncation, resummation, or numerical methods. The choice of method depends on the specific problem and the desired level of accuracy.

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