Complex output from a real integral

[member 428835]

Hi PF!

I am integrating the following

sin\[Theta][x_, \[Alpha]_] := Sqrt[(2 - 2 x^2)/(
 3 - 4 x Cos[\[Alpha]] + Cos[2 \[Alpha]])]

cos\[Theta][x_, \[Alpha]_] := Sqrt[(Cot[\[Alpha]] - x Csc[\[Alpha]])/(
 1 + 2 x Cot[\[Alpha]] Csc[\[Alpha]] - 2 Csc[\[Alpha]]^2)]

\[Rho][x_, \[Alpha]_] := Sqrt[(3 - 4 x Cos[\[Alpha]] + 
    Cos[2 \[Alpha]]) Csc[\[Alpha]]^2]/Sqrt[2]

dn\[Phi]b[j_, x_, \[Alpha]_, L_] := 
 j LegendreP[j, L, 
     cos\[Theta][
      x, \[Alpha]]] (-Sqrt[1 - x^2] sin\[Theta][x, \[Alpha]] + 
      x cos\[Theta][x, \[Alpha]]) \[Rho][x, \[Alpha]]^(j - 1) + 
   sin\[Theta][x, \[Alpha]] D[
     LegendreP[j, L, z], {z, 
      1}] (Sqrt[1 - x^2] cos\[Theta][x, \[Alpha]] + 
      x sin\[Theta][x, \[Alpha]]) \[Rho][x, \[Alpha]]^(j - 1) /. 
  z -> cos\[Theta][x, \[Alpha]]

b[\[Alpha]_] := Cos[\[Alpha]]

NIntegrate[dn\[Phi]b[3, x, \[Pi]/3, 1], {x, b[\[Pi]/3], 1}]

The output for this integral is $-1.1331 - 1.17012 i$. A plot of the integrand is attached. There is a singularity, but the imaginary component seems wrong. Any help?

 

[JMz]

1. If you would like assistance, it would help if you converted as much as you can of that code into conventional algebraic & analysis symbolic expressions that don't require "\special symbols", language-specific syntax, and (I imagine) version-specific built-in or special functions.
2. If that integrand is giving you a non-Real result, then presumably your integration path is responsible. But I have no idea where to find that specified within the code (because I don't happen to be fluent in that language).

 

[DrClaude]

1. If you would like assistance, it would help if you converted as much as you can of that code into conventional algebraic & analysis symbolic expressions that don't require "\special symbols", language-specific syntax, and (I imagine) version-specific built-in or special functions.

I'll temper that by saying that having code that can be copied and directly pasted into Mathematica is very useful when trying to help on the technical aspects of Mathematica.

However, @joshmccraney, I think that the problem here is not Mathematica but mathematics. You should start a new thread in the calculus forum with a mathematical description of the integral.

 

[JMz]

My apologies, @joshmccraney, for failing to notice that the thread's keyword is, in fact, Mathematica. (I suspected as much but failed to recognize that expressing the OP in Mathematica with that keyword was entirely appropriate -- even though the problem does indeed seem to lie within math, not Math.)

 

[member 428835]

Thank you both. I'll create another thread, as it was unclear to me whether Mathematica was the culprit or the actual math.

 

[JMz]

Sounds good. (Great thumbnail, BTW. ;-)

 

[member 428835]

Thanks! I'm a huge fan!

 

[JMz]

(Me, too.)