# Series expansion of a given function

I attempt to do the series expansion for the following function

Simplify[Series[((-5)^(1/
6) (2 + (-2 + ξ3) ξ3 + (-2 + ξ4) ξ4) a[
1])/((-1 + ξ3)^2 ξ3^4 (-6 + 5 ξ3)^(4/
5) (-1 + ξ4)^2 ξ4 (-6 + 5 ξ4)^(1/5)), {ξ3, 0,
1}], Im[ξ3] == 0]


Note that this function is expected to expand this function to order $$\xi_3$$. However, it fails to capture order $$\frac{1}{\xi_3}$$ term

If I slightly modify the code to expand the function to order $$\xi^2$$, then I could get the correct order $$\frac{1}{\xi_3}$$ term

• I'm getting the same result for each. One is perhaps less simplified than the other, but they are equivalent. Commented May 31 at 18:32
• @DanielLichtblau I attached my mma result. Maybe it's because my version is a bit old? It's 12 Commented Jun 3 at 6:55
• It is possible this improved in the last few releases. I know some issues with getting correct order for Series were addressed in this time frame. Commented Jun 3 at 17:02
• @DanielLichtblau Thanks for this info. Perhaps the best way is to update my Mathematica... Commented Jun 4 at 0:19

I am not sure why this is happening, but here's a quick fix. Do a replacement $$\xi_3 \rightarrow \tfrac{1}{x_3}$$ and then expand around $$\infty$$.

expr = ((-5)^(1/
6) (2 + (-2 + ξ3) ξ3 + (-2 + ξ4) ξ4) a[
1])/((-1 + ξ3)^2 ξ3^4 (-6 + 5 ξ3)^(4/
5) (-1 + ξ4)^2 ξ4 (-6 + 5 ξ4)^(1/5));
expr /. ξ3 :> 1/xx3;
Series[%, {xx3, Infinity, 1}]
Coefficient[%, xx3]


Use Assuming.

Assuming[\[Xi]3 \[Element] Reals,
Series[((-5)^(1/
6)  (2 + (-2 + \[Xi]3)  \[Xi]3 + (-2 + \[Xi]4)  \[Xi]4)  a[
1])/((-1 + \[Xi]3)^2  \[Xi]3^4  (-6 + 5  \[Xi]3)^(4/
5)  (-1 + \[Xi]4)^2  \[Xi]4  (-6 + 5  \[Xi]4)^(1/
5)), {\[Xi]3, 0, 1}] // Simplify]


• Do you know why does Assuming help here? Commented May 30 at 14:27
• @Vayne In your code, you told Simplify that Im[ξ3] == 0, but you didn't tell Series that. Assuming let both of them know that. Commented May 30 at 15:09
• Thanks! But why does the additional assumption could help series expansion capture 1/\xi term? Commented May 31 at 5:05