- #1
member 428835
Hi PF!
I am integrating the following
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?
I am integrating the following
Code:
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}]