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527form1

# 527form1 - 642:527 FORMULA SHEET FOR EXAM 1 FALL 2009...

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Unformatted text preview: 642:527 FORMULA SHEET FOR EXAM 1 FALL 2009 Taylor series (with radii of convergence given): 1 = 1 + x + x2 + Â· Â· Â· = 1âˆ’x x âˆž xn , 0 âˆž |x| < 1 xn , n! |x| < âˆž |x| < âˆž |x| < âˆž x2 x3 e =1+x+ + + Â·Â·Â· = 2! 3! cos x = 1 âˆ’ sin x = x âˆ’ x4 x2 + âˆ’ Â·Â·Â· = 2! 4! x3 x5 + âˆ’ Â·Â·Â· = 3! 5! âˆž âˆž 0 0 âˆž (âˆ’1)n x2n , (2n)! (âˆ’1)n x2n+1 , (2n + 1)! 0 The Gamma function. For x > 0, Î“(x) = 0 txâˆ’1 eâˆ’t dt. If x is not 0 or a negative integer, Î“(x + 1) = xÎ“(x). âˆš If n is a non-negative integer, Î“(n + 1) = n!. Î“(1/2) = Ï€ . âˆž The Method of Frobeniousâ€”solution forms: âˆž y1 (x) = xr n=0 an xn (= y2 (x) ?), âˆž y2 (x) = y1 (x)(ln x) + x r1 n=1 bn x , n y2 (x) = Cy1 (x)(ln x) + x r2 n=0 bn xn . The Method of Frobeniousâ€”useful formula: uâ€²â€² + p(x)uâ€² + q (x)u = Bessel Functions. A. The Bessel equation of order Î½ : B. Bessel functions: JÎ½ (x) = Jâˆ’Î½ (x) = x 2 x 2 Î½âˆž C â€² [y1 (x) âˆ’ xp(x)y1 (x) âˆ’ 2xy1 (x)] . x2 x2 y â€²â€² + xy â€² + (x2 âˆ’ Î½ 2 )y = 0. k =0 âˆ’Î½ âˆž (âˆ’1)k k!Î“(Î½ + k + 1) (âˆ’1)k k!Î“(k âˆ’ Î½ + 1) x 2 x 2 2k 2k k =0 (cos Î½Ï€ )JÎ½ (x) âˆ’ Jâˆ’Î½ (x) , if v = 0, 1, 2, . . . . YÎ½ (x) = sin Î½Ï€ Yn (x) = lim YÎ½ (x), if n = 0, 1, 2, . . . . Î½ â†’n On the exam, the remaining part of this formula sheet will be Appendix C, the Laplace transform tables, from the text. ...
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