Legendre Polynomials and Complex Analysis

["pi"mp]

Hi all,

I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre functions which I know pop up in the angular solutions to Schrödinger's equation in spherical coordinates.

Then I had an idea that may turn out to be very naive:

I know that complex analysis at times, reduces a contour integral to a theory of the behavior of functions over the complex plane (Cauchy-Goursatt Thm., Residue theory, etc.) so I just looked at the Legendre function as a function of z and its consequences. My real hope lies in the fact that being an angular solution, they depend only on $$\theta$$ and $$\phi$$. So my idea was to curl the complex plane up into a sphere (stereographic projection) so that I can call any complex number $$\zeta$$ by calling a given $$\theta$$ and $$\phi$$. Then I can put the Associated Legendre polynomials in terms of theta and phi and sort of picture it as "living on the sphere."

I am hoping that I can maybe use residue theory or some other theory in complex analysis to evaluate integrals in quantum mechanics. I am just wondering if this seems hopeful, is already well-established, or if it seems like a naive, hopeless idea.

Thanks a lot

 

[TriTertButoxy]

Do you mean $\zeta=\theta + i\phi$? or are you thinking of a more complicated map?

I think your idea is rather intriguing, but I'm worried that since the Legendre polynomials are analytic (provided $\theta$ and $\phi$ are real), I'm afraid there's not too much you might be able to do with them.

My experience with complex analysis is that you want functions with interesting analytic structure (poles/branch points) to be useful. You may want to consider analytically continuing $\theta$ and $\phi$ into the complex plane as well.

 

[Bill_K]

Associated Legendre functions do not depend on φ. They're a function of Θ only.

 

["pi"mp]

I think your idea is rather intriguing, but I'm worried that since the Legendre polynomials are analytic (provided $\theta$ and $\phi$ are real), I'm afraid there's not too much you might be able to do with them.

yes exactly, although the Associated Legendre polynomials will have at least one isolated singularity I believe so I was hoping to apply residue theory perhaps.

Bill, do you mind elaborating a bit. I don't quite understand why. Thank you.

oh and the map I'm using is $$\zeta= cot((phi/2))e^itheta$$

 

[Bill_K]

The associated Legendre function is

P_ℓm(x) = (-1)^m/2^ℓℓ! (1-x²)^m/2 d^ℓ+m/dx^ℓ+m(x²-1)^ℓ

where x = cos θ. No φ dependence means it can't be written as a function of ζ. You can consider making x complex, but that's not the same as stereographic coordinates.

 

["pi"mp]

ah okay I see what you mean...but what I did with the regular legendre function was put it in terms of z. And since the stereographic projection is a bijection from the plane to the sphere (minus the point at infinity), I can call any z by calling the corresponding theta and phi on the sphere.

Why will this not also work for the Associated Legendre function? Because clearly, when m is negative, inputting 1 into the function is undefined so I was hoping to maybe apply residue theory perhaps?

I understand what your argument is, just not why my process doesn't work also. Thanks

 

[Hurkyl]

P.S. for math that's supposed to be inlined in your paragraph, use [ itex ] instead of [ tex ]

 

["pi"mp]

ah thanks, I'm still not great with LaTex