Integrating the Gegenbauer function (1st kind) wrt theta

[Ariana1983]

Homework Statement

Find ∫ I_n(ζ) dθ , where ζ= cosθ and I_n(ζ) is the gegenbauer function of the first kind.

The original problem is to find find ∫ sinθ *I^//_n(ζ) dθ where I^//n= 2nd differential of I_n(ζ) with respect to ζ.

Homework Equations

d/dζ I_n(ζ)=-P_n-1(ζ), where P_n(ζ) is the legendre function of the first kind.

I_n(ζ)= [P_n-2(ζ)-P_n(ζ)]/(2n-1)

The Attempt at a Solution

I know d/dθ= d/dζ * dζ/dθ = -sinθ*d/dζ by the chain rule since ζ=cosθ, but how to apply that to integration of the gegenbauer with respect to theta? I came across that problem as I used integration via parts to solve the original problem.

 

[mighty2000]

To solve this problem, we can use the substitution method. Let u = In(ζ) and du = d/dζ In(ζ) dζ. Then, using the chain rule, we have d/dθ In(ζ) = d/dζ In(ζ) * dζ/dθ = -sinθ * d/dζ In(ζ) = -sinθ * du.

Substituting this into the original integral, we have ∫ -sinθ * In(ζ) dθ = ∫ u du.

Integrating both sides with respect to θ, we get ∫ -sinθ * In(ζ) dθ = ∫ u du = u + C = In(ζ) + C.

Therefore, the final solution is ∫ In(ζ) dθ = In(ζ) + C, where C is a constant of integration.