extra-credit - x ). But how to memorize f ( x, z )?...

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Extra credit problem * Due date: Anytime before December 1st, 2007 The Legendre polynomials are coefficients of z n in a Taylor series expension with respect to z of a certain function f ( x, z ): f ( x, z ) = X n =0 P n ( x ) z n , | x | , | z | < 1 . This function is called the generating function for the Legendre polynomials. The purpose of this problem is to show that this function is: f ( x, z ) = 1 1 - 2 xz + z 2 . To show this, you should only use the following recurrence relation that exists between Legendre polynomials (no need to prove this recurrence relation): ( n + 1) P n +1 ( x ) = (2 n + 1) xP n ( x ) - nP n - 1 ( x ) , n = 0 , 1 , . . . and the fact that P 0 ( x ) = 1. Follow the procedure outlined in problem 35 of section 8.8. Practical use : To write down the Legendre polynomials explicitely without memorizing them, it suffices to expand the function f ( x, z ) in a Taylor series with respect to z (while fixing
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Unformatted text preview: x ). But how to memorize f ( x, z )? Here’s a graphical way with an interesting physical interpretation. Let an electrical charge q be located at a point with polar coordinates ( r, θ ) (here, θ is the angle between the radius and the y-axis) with r < 1. In physics one shows that the potential at the point with polar coordinates (1 , 0) on the y-axis is proportional to 1 /r where r is the distance between this point and the point where the charge is: potential ∼ 1 r . * MAP 4305; Instructor: Patrick De Leenheer. 1 q r r ’ theta 1 Now by the law of cosines: ( r ) 2 = 1 + r 2-2 r cos( θ ) , and thus the potential is proportional to: 1 p 1 + r 2-2 r cos( θ ) , which is exactly f (cos( θ ) , r ). 2...
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.

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extra-credit - x ). But how to memorize f ( x, z )?...

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