# Is this definite integral solvable?

GROUPS:
 I have been looking at the eccentricity in induction motor. Through using the Maxwell's equations I have arrived at the equation (1) in the attached file.However, I cannot find a solution for the equation (2) in a closed form using Mathematica. I believe, that the solution can be greatly simplified as described in the attached file. Am I correct or not? Attachments:
1 year ago
8 Replies
 Kay Herbert 1 Vote By any chance, is ε a mall parameter ε<<1 ?
1 year ago
 Hello Key Herbert, The eccentricity in the induction motor can be all the way to ε=1. Of course it cannot be exactly 1, because it would mean that rotor is rubbing again the startor a disaster for any electric motor. The eccentricity is kind of misterious, because lots of people talk about it, but nobody wants to actually measure it. From my experience, the eccentricity is usually below 0.3. Hence the approximate solution would not give any meaningful solution. Thank you for the response, I have never hoped for something so fast. Regards, jank
1 year ago
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1 year ago
 Daniel Lichtblau 1 Vote If you post actual Mathematica code people are more likely to try it out.
1 year ago
 Kay Herbert 1 Vote If I use symmetry take 2 x integral from 0 to Pi and define the constraints on the parameters, Mathematica gives me an analytical answer to the integral: In[34]:= Integrate[2 Cos[p b]/ (1 + e Cos[b]), {b, 0, Pi}, Assumptions -> e > 0 && p > 0 && p \[Element] Integers && e <= 1] Out[34]= -(( 2 \[Pi] HypergeometricPFQRegularized[{1/2, 1, 1}, {1 - p, 1 + p}, ( 2 e)/(-1 + e)])/(-1 + e)) 
 Kay Herbert 1 Vote Hyper geometric functions aren't uncommon. if p=2 it greatly simplifies: In[19]:= -(( 2 \[Pi] HypergeometricPFQRegularized[{1/2, 1, 1}, {1 - p, 1 + p}, ( 2 e)/(-1 + e)])/(-1 + e)) /. p -> 2 Out[19]= (2 (-2 + 2 Sqrt[(-1 - e)/(-1 + e)] - 2 Sqrt[(-1 - e)/(-1 + e)] e + e^2) \[Pi])/(Sqrt[(-1 - e)/(-1 + e)] (-1 + e) e^2) In[20]:= Simplify[( 2 (-2 + 2 Sqrt[(-1 - e)/(-1 + e)] - 2 Sqrt[(-1 - e)/(-1 + e)] e + e^2) \[Pi])/(Sqrt[(-1 - e)/(-1 + e)] (-1 + e) e^2)] Out[20]= -(( 2 (2 - e^2 - 2 Sqrt[(1 + e)/(1 - e)] + 2 e Sqrt[(1 + e)/(1 - e)]) \[Pi])/((-1 + e) e^2 Sqrt[(1 + e)/( 1 - e)]))