Series vs Asymptotic in 12.1

Question

The functionality of Series and Asymptotic (new in V12.1) is very similar. In fact, they are both listed in the Asymptotics guide page. That same guide page hints that Asymptotic is more general than Series: while Series is just for functions, Asymptotic can handle integral transformations among other things.

This begs the question of when should one ever use Series if Asymptotic can apparently do everything Series could before? I remember this same question being raised during the development of V12.1, but I don't think it was ever really answered (term ordering and "propagation" of accuracy were mentioned). Now that V12.1 is out, has that changed? Is there something Series is better at than Asymptotic, or is Asymptotic going to supersede Series, which is now being kept for backwards compatibility only?

Answer 1

Extended comment, I won't accept this as an answer.

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Here are some cases I've found where Series might be a better choice than Asymptotic. This list doesn't show when Asymptotic could be better, since those cases are fairly well documented.

• Series returns a SeriesData object, while Asymptotic returns a simple expression. Because of this, computations with the generated series is significantly easier with Series.

• As an example of the previous point, exponentiating a series expansion gives more sensible results when using Series:

  Exp[Series[Sin[x], {x, 0, 5}]]
  (* 1 + x + x^2/2 - x^4/8 - x^5/15 + O(x^6) *)
  
  Exp[Asymptotic[Sin[x], {x, 0, 5}]]
  (* Exp[x - x^3/6 + x^5/120] *)
  

• Similarly, arithmetic operations with SeriesData automatically cuts off higher-order terms when needed. If I add two series, where one has five terms and the other only two, the resulting sum is also only second-degree, which is not what happens with Asymptotic:

  Series[Sin[x], {x, 0, 5}] + Series[Cos[x], {x, 0, 2}]
  (* 1 + x - x^2/2 + O(x^3) *)
  
  Asymptotic[Sin[x], {x, 0, 5}] + Asymptotic[Cos[x], {x, 0, 2}]
  (* 1 + x - x^2/2 - x^3/6 + x^5/120 *)
  

• Two series with different expansion centers cannot be combined with Series:

  Series[Exp[x], {x, 0, 3}] + Series[Exp[x], {x, 1, 3}]
  (* Series in x to be combined have unequal expansion points 0 and 1 *)
  

  Asymptotic will, for better or for worse, happily combine them:

  Asymptotic[Exp[x], {x, 0, 3}] + Asymptotic[Exp[x], {x, 1, 3}]
  (* 1 + E/3 + x + (E x)/2 + x^2/2 + x^3/6 + (E x^3)/6 *)
  

• Series supports "multivariate" series expansions out-of-box, while Asymptotic does not:

  Series[Sin[x + y], {x, 0, 2}, {y, 0, 2}]
  (* (y + O(y^3)) + x(1 - y^2/2 + O(y^3)) + x^2(-(y/2) + O(y^3)) + O(x^3) *)
  
  Asymptotic[Asymptotic[Sin[x + y], {x, 0, 2}], {y, 0, 2}]
  (* x + y - (x^2 y)/2 - (x y^2)/2 *)
  

• Asymptotic is significantly slower for hypergeometric functions (and possibly other special functions):

  Series[Hypergeometric2F1[a, b, c, x], {x, Infinity, 1}] // AbsoluteTiming
  (* {0.030965, ...} *)
  
  Asymptotic[Hypergeometric2F1[a, b, c, x], {x, Infinity, 1}] // AbsoluteTiming
  (* {15.3086, ...} *)
  

  This may be a bug, as running Series with the new input notation x->Infinity

  Series[Hypergeometric2F1[a, b, c, x], x -> Infinity]
  

  runs for more than two minutes (didn't wait for completion, reported to WRI).