# Question:Discrepancy between numerical and analytical results for elliptic integrals

## Question:Discrepancy between numerical and analytical results for elliptic integrals

Maple

As usual, I have a tricky question. There is an integral that Maple can take numerically

R0 := 1/(a-sqrt(b+c*cos(x)));

Now let's put the coefficients, e.g.

a := 0.9; b := 4.5; c :=0.1

and take the integral from 0 to 2*Pi

R1 := evalf(int(R0, x = 0 .. 2*Pi));

Also, there is an exact analytical result that Maple gives (I give it after simplifying it to avoid division by zero for the limit x=0 and x=2*Pi)

R2:=-4*((a^2-b+c)*EllipticK(sqrt(-2*c/(b-c)))-a^2*EllipticPi(2*c/(a^2-b+c), sqrt(-2*c/(b-c))))/((a^2-b+c)*sqrt(b-c));

As it turns out, the results are completely different. In the first case -5.145818656, while for the second case -3.612771378+0.I

Moreover, If we change the coefficients to a := 0.9; b := 4.5; c := -4 then I obtain Float(undefined)+3.662506136*I and -2.362349457+3.662506117*I , respectively.

My question: how to avoid this descepancy?

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