Elliptic Integral between Zero and 2 Pi has no result?

[nassemabdullah (ali kreem)]

Hello,

Please I am trying to solve Elliptic Integral for period [0-2PI]. I got the result for unlimited integral but when I apply limited integral I got nothing

For limited integral I used below equation and it is not get through

WolframAlpha["Integrate[Cos x/[R^2+r^2-[2*R*r*Cos x]+[k-k1]^2]^1/2],{x,0,2\[Pi]}"]

For unlimited integral I use below equation and it is work

WolframAlpha["Integrate[Cos x/[R^2+r^2-[2*R*r*Cos x]+[k-k1]^2]^1/2]}"] 

Thanks

[jofreep (Jofre Espigule-Pons)]

Integrate[Cos x/(R^2 + r^2 - (2Rr*Cos x) + (k - k1)^2)^1/2, {x, 0, 2 [Pi]}]

Evaluating the integral in Mathematica yields the following result:

ConditionalExpression[(-4 Cos \[Pi] r R + ((k - k1)^2 + r^2 + R^2) (Log[(k - k1)^2 + r^2 + R^2] - 
         Log[(k - k1)^2 + r^2 - 4 Cos \[Pi] r R + R^2]))/(8 Cos r^2 R^2), 
     Re[((k - k1)^2 + r^2 + R^2)/(Cos r R)] > 4 \[Pi] || 
      Re[((k - k1)^2 + r^2 + R^2)/(Cos r R)] < 
       0 || ((k - k1)^2 + r^2 + R^2)/(Cos r R) \[NotElement] Reals]

[nassemabdullah (ali kreem)]

Hello,

Thank you so much for your reply. I made mistake by typing. I mean the interval is [0-2PI]. After I fixed it I still not get result.

Any help will be very appreciate.

Thanks

[mthiel (Marco Thiel)]

Dear Jofre,

are you sure that your syntax is ok? In your expression

Integrate[Cos x/(R^2 + r^2 - (2*R*r*Cos x) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]

the Cos has no proper argument - there are no square brackets. And there is a difference to that and Cos[x]:

If I use

Integrate[Cos[x]/(R^2 + r^2 - (2*R*r*Cos[x]) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]

the result is somewhat different.

I suppose that the original question is about WolframAlpha, where none of this matters, because it does not evaluate.

As a suggestion to Ali Kreem I would use the free tier of the WolframCloud. It allows you to use proper Mathematica syntax and your integral will evaluate.

Cheers,

Marco

[nassemabdullah (ali kreem)]

Hello,

Thank you so much for your reply. Please below the result I got for unlimited integral. but for integral with limits [0-2PI]there is no answer. any help will be very appreciate.

Regards

[mthiel (Marco Thiel)]

It appears that you have access to the Wolfram Language directly (for example you are using Mathematica). In that case you should not make a WolframAlpha request at all. As I wrote in my last post

Integrate[Cos[x]/(R^2 + r^2 - (2*R*r*Cos[x]) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]

should help. It appears that you are only interested in real solutions. So

Assuming[x \[Element] Reals && r \[Element] Reals && R \[Element] Reals && k \[Element] Reals && k1 \[Element] Reals, 
 Integrate[Cos[x]/(R^2 + r^2 - (2*R*r*Cos[x]) + (k - k1)^2)^1/2, {x, 0, 2 \[Pi]}]]

makes that slightly more explicit. This gives

or in traditional form

Cheers,

Marco

[nassemabdullah (ali kreem)]

Hello,

Thank you so much. It works

Kind Regards

[jofreep (Jofre Espigule-Pons)]

Thanks Marco! Somehow I copy-pasted the original input without even realizing the cosines without argument and I gave its result without thinking.