# Question:A series of expansion of incomplete elliptic integrals for a complex argument and numerical results

## Question:A series of expansion of incomplete elliptic integrals for a complex argument and numerical results

Maple 17

Good day.

I have a problem with the numerical approximation of incomplete elliptic integrals of the first kind in the case when the argument and the modulus are complex numbers.

Let's consider an incomplete elliptic integral of the first kind EllipticF((a+I*b)*x, c+I*d), where a,b,c,d are arbitrary real numbers and x is very large number. Here is the code:

```restart;
R0 := EllipticF((a+I*b)*x, c+I*d);
with(MultiSeries, series):
R1 := series(R0, x = infinity, 2);
a := .5; b := 1.75; c := 10; d := 12.5; x := 10^6;
evalf(R0); evalf(R1);```

Below you can see that answers are different

```.1680611942+.2058775337*I
-0.4563467782e-1-.3592094173*I+O(1/1000000000000000000)```

While the same code for an incomplete elliptic integral of the second kind gives the correct approximation:

```restart;
R00 := EllipticE((a+I*b)*x, c+I*d);
R11 := series(R00, x = infinity, 2);
a := .5; b := 1.75; c := 10; d := 12.5; x := 10^6;
evalf(R00); evalf(R11);
-1.687498740*10^7+2.374999011*10^7*I
-1.687501260*10^7+2.375000989*10^7*I+O(1/1000000)```

What is the difference?

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