New Asymptotic function in Mathematica 12.1

One of the new features in Mathematica 12.1 is the function Asymptotic. Here’s a quick example of using it.

Here’s an asymptotic series for the log of the gamma function I wrote about here.

\log \Gamma(z) \sim (z - \frac{1}{2}) \log z - z + \frac{1}{2} \log(2\pi) + \frac{1}{12z} - \frac{1}{360z^3} + \cdots

If we ask Mathematica

    Asymptotic[LogGamma[z], z -> Infinity]

we get simply the first term:

z Log[z]

But we can set the argument SeriesTermGoal to tell it we’d like more terms. For example

    Asymptotic[LogGamma[z], z -> Infinity, SeriesTermGoal -> 4]

yields
-1/(360*z^3) + 1/(12*z) - z + Log[2*Pi]/2 - Log[z]/2 + z*Log[z]

This doesn’t contain a term 1/z4, but it doesn’t need to: there is no such term in the asymptotic expansion, so it is giving us the terms up to order 4, it’s just that the coefficient of the 1/z4 term is zero.

If we ask for terms up to order 5

    Asymptotic[LogGamma[z], z -> Infinity, SeriesTermGoal -> 5]

we do get a term 1/z5, but notice there is no 4th order term.

1/(1260*z^5) - 1/(360*z^3) + 1/(12*z) - z + Log[2*Pi]/2 - Log[z]/2 + z*Log[z]

A note on output forms

The Mathematica output displayed above was created by using

Export[filename, expression]

to save images as SVG files. The alt text for the images was created using

InputForm[expression].

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