Chapter 5, PROBLEM SET 5.2 #10

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Generating functions play a significant role in modern applied mathematics. The idea is simple. If we want to study a certain sequence (f_{n}(x)) and can find a function

we may obtain properties of (f_{n}(x)) from those of G, which €œgenerates€ this sequence and is called a generating function of the sequence.

(a) Show that

is a generating function of the Legendre polynomials. Start from the binomial expansion of 1/ˆš1-v, then set v = 2xu - u^{2}, multiply the powers of 2xu - u^{2 }out, collect all the terms involving u^{n}, and verify that the sum of these terms is p_{n}(x)u^{n}.

(b) Let A_{1} and A_{2} be two points in space (Fig. 108, r_{2} > 0). Using (12), show that

This formula has applications in potential theory. (Q/r is the electrostatic potential at A_{2} due to a charge Q located at A_{1}. And the series expresses 1/r in terms of the distances of A_{1 }and A_{2 }from any origin O and the angle Î¸ between the segments OA_{1} and OA_{2}.)

(c) Show that P_{n}(1) = 1, P_{n}(-1) = (-1)^{n}, P_{2n+1}(0) = 0, and P_{2n}(0) = (-1)^{n} · 1 · 3 . . . (2n - 1)/[2 · 4 . . .(2n)].

PROBLEM SET 5.2:

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PROBLEM SET 5.4:

PROBLEM SET 5.5: