I just returned home, pulled out my ancient copy of Whittaker and Watson, and now see what the issue is. In the documentation for EllipticE in Mathematica one sees that it is defined as

Integrate[(1 - m Sin[t]^2)^(1/2), {t 0, Pi/2}]

whereas other definitions use m^2 as in

Integrate[(1 - m^2 Sin[t]^2)^(1/2), {t 0, Pi/2}]

and this will account for the factor of 1/2 difference between the usages.

So everyone is correct but the definitions of how the modulus parameter is used in the definition differ.

Hello,

Thank you so much for your reply.

For this code. I am trying to find partial derivative of U with respect to y. I have k^2 = ((4*R*r)/((y - Y)^2 + (r + R)^2)).

My question is.

Do I have to define k= ((4Rr)/((y - Y)^2 + (r + R)^2))or k=((4Rr)/((y - Y)^2 + (r + R)^2))^1/2`

In order to get denominator without 2? In other words how I can define k to get answer without 2? I still got answer with 2 even when I define k in both cases.

Clear[r, R, y, Y, k, U, Ur, Uy]
k = ((4*R*r)/((y - Y)^2 + (r + R)^2))^1/2;
U = EllipticE[k];
Uy = D[U, y]

The result

-(((y - Y) (EllipticE[(2 r R)/((r + R)^2 + (y - Y)^2)] - 
    EllipticK[(2 r R)/((r + R)^2 + (y - Y)^2)]))/((r + R)^2 + (y - 
    Y)^2))

Hello,

Thank you for your reply. Really I do not know if the correct answer is with 2 in denominator or no. I am confused now.

I will wait to your reply.

Regards

Hello,

Thank you for your reply. It is not the same result.

Thanks