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.

If we ask Mathematica

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

we get simply the first term:

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

This doesn’t contain a term 1/*z*^{4}, 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/*z*^{4} 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/*z*^{5}, but notice there is no 4th order term.

## 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*`]`

.