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Topic: Legendre Functions at Large Order and Argument>1 ?
Replies: 2   Last Post: Jul 1, 1996 3:50 AM

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 AES Posts: 72 Registered: 12/7/04
Legendre Functions at Large Order and Argument>1 ?
Posted: Jun 29, 1996 4:16 PM

I'd very much like to find asymptotic expressions for Legendre functions
and associated Legendre functions in the limit of very large order and for
real arguments _greater_ than unity (in other words, outside their usual
range).

A physical problem leads to the expression (in Mathematica notation)

c[n_,theta_] = Sqrt[n! LegendreP[n, 1/Cos[2 theta]] / Cos[2 theta]] *
LegendreP[n/2, -n/2, 1/Cos[2 theta]]

Numerical evaluation shows that as n becomes very large this expression
increases in magnitude with increasing n for theta > thetaC; appears to
approach a constant value for theta = thetaC; and decreases with
increasing n for theta < thetaC; where the critical value thetaC is very
close to (equal to?) Pi/12. But the numerical evaluation runs out of
steam for orders greater than n = 60 or so.

Pointers much appreciated. siegman@ee.stanford.edu

(The "ee.stanford" gives it away; I know enough math to know what a
Legendre function is, but not enough to easily deal with its asymptotic
values.)

Date Subject Author
6/29/96 AES
6/29/96 meron@cars3.uchicago.edu
7/1/96 Paul Abbott