## Expansion of $E(i c \mid m)$ at $c\to\infty$?

5

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result given by Mathematica 9 on Linux (which I was using previously).

In Mathematica 8 on Windows the following input:

Series[EllipticE[I c, m], {c, Infinity, 0}] // PowerExpand // FullSimplify


produces a warning:

General::ivar: I c is not a valid variable. >>

and the following output (slightly rearranged by me to better fit the browser):

$$-\frac{i\left(6m+\frac{(1+m)}{\sinh^2(c)}\right)}{6\sinh(c)m^{3/2}}-\text{EllipticE}[m]+\frac{m \text{EllipticE}[\frac{1}{m}] + (1-m)\text{EllipticK}[\frac{1}{m}] }{\sqrt{m}}-$$ $$\frac{(1+m \cosh(2 c))\sqrt{\text{Limit}\big[-m\sinh^2(c)~,~i c \to 0\big]}}{\sinh^2(c)2m}$$

Same output as a code:

-((I (6 m Csch[c] + (1 + m) Csch[c]^3))/(6 m^(3/2))) -
EllipticE[m] +
(m EllipticE[1/m] - (-1 + m) EllipticK[1/m])/Sqrt[m] -
(((1 + m Cosh[2 c]) Csch[c]^2 Sqrt[Limit[-m Sinh[c]^2, I c -> 0]])/(2 m))


Now, in Mathematica 9 on Linux there was no such warning, and no Limit term appeared. I am confused about how to treat this Limit term, since it might just be a sign of something going terribly wrong in the guts of Mathematica 8. Does anyone have an advice on how to proceed? Maybe some of you can evaluate the same series expansion in a different version of Mathematica so that we could compare results?

### EDIT

Series[EllipticE[c, m], {c, I Infinity, 0}] // PowerExpand // FullSimplify

1From a strictly mathematical point of view, Limit[-m Sinh[c]^2, I c -> 0] should be zero... – Federico – 2013-03-23T17:05:22.310