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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.)



