Asymptotic Expansions for Some Special Functions

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 .

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