Asymptotic Expansions for Some Special Functions

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The most common type of asymptotic expansion for a function is a formal series that can be truncated after a finite number of terms to a sum that provides an approximation to the function for large values of . This is usually written as



Most often, the series diverges for any fixed . But for fixed , the truncated sum approaches the function as —that is,


Sometimes fewer terms can be used when the function is represented in the form

where is the leading term of the asymptotic expansion.

A well-known example is Stirling's asymptotic series for the gamma function:


A common way of generating an asymptotic expansion is to apply repeated integration by parts, beginning with an integral representation for . Another approach makes use of Laplace's method of steepest descent.

This Demonstration shows an easier way to derive asymptotic expansions, using the capability of Mathematica to compute a power series for a function about the point using Series[f[x],{x,∞,n}]. The function can be determined using Series[f[x],{x,∞,0}].

You can check the numerical accuracy of the asymptotic series. The fractional error is plotted for values of .


Contributed by: S. M. Blinder (August 2018)
Open content licensed under CC BY-NC-SA


An asymptotic series is divergent if has an essential singularity at . The series is said to represent asymptotically if



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