# Question:trouble with MultiSeries: series resp asympt for EllipticF

## Question:trouble with MultiSeries: series resp asympt for EllipticF

Maple
```I want the asymptotics for EllipticF(z,k) w.r.t. z (yes, not k).

My example is F:=I*EllipticF(c*I/x,k), c=2-2^(1/2),k=3+2*2^(1/2)
and I want the limit x ---> 0+.

MultiSeries:-limit(E,x=0);

results in 0.54... + 0.54...*I (real = imag, positive value).

MultiSeries:-series(E,x,3);

gives (after clearing the output) the same for setting x=0.

For Numerics my setting is Digits:=24 and eps:=1/10^18.

subs(c=2-2^(1/2), k=3+2*2^(1/2), F);
eval(%, x=eps); evalf(%);

-0.54304590015531201290853733343625 - 0. I

which is a negative Real.

For cross checking write as integral and make an allowed
change of coordinates:

convert(F,Int): subs(_alpha1=eta,%): combine(%);
Change(%,eta*I/c = -xi,xi);

1/x
/
|                     1
-c  |     ---------------------------------- dxi
|            2  2 1/2       2   2  2 1/2
/      (1 + xi  c )    (1 + k  xi  c )
0

This shows, that we integrate over a positive integrand and
finally multiply by -c ~ -0.58 ( and continuing by value(%);
simplify(%) assuming 0 < c; gives back the F).

So MultiSeries result can not be correct.

Any idea?
```
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