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Appendix II I« Nvunerlcal Evaluation of Some 

 Definite Integrals 



In part III, the problem of evaluating the definite 

 Integrals In equations (3.7), (3. 11) arose. These Integrals 

 have square root singularities at the limits of Integration. 

 In this appendix we shall develop a quadrature formula, which 

 was used for this evaluation but does not seem to be as well- 

 known as It should. It is based on Tchebycheff polynomials. 



The nth Tchebycheff polynomial T (x), -1 ^ x ^ 1, is 

 defined by 



T (x) = cos(n arc cos x), 



or in other terms, T (x) = cos n© , where x = cos 6. 



The Tchebycheff polynomials are orthogonal with respect to 

 the weight function , 1. e., 



T„(x) T^(x) 



dx = If n ^ m. 



-I 



^ 



^2 



This follows immediately when we make the transformation 

 X = cos 9. 



By virtue of the orthogonality property, the follow- 

 ing theorem can be proved: 



Theorem . Let x,, x-,..., x^ be the n zeros of 



T (x) in the range -1 ^ x ^ 1. Let f (x) be any polynomial 

 of degree at most 2n-l. Then 



.1 n 



1 





v., ^ 



