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 + · · · = 1−x x ∞ xn , 0 ∞ |x| < 1 xn , n! |x| < ∞ |x| < ∞ |x| < ∞ x2 x3 e =1+x+ + + ··· = 2! 3! cos x = 1 − sin x = x − x4 x2 + − ··· = 2! 4! x3 x5 + − ··· = 3! 5! ∞ ∞ 0 0 ∞ (−1)n x2n , (2n)! (−1)n x2n+1 , (2n + 1)! 0 The Gamma function. For x > 0, Γ(x) = 0 tx−1 e−t dt. If x is not 0 or a negative integer, Γ(x + 1) = xΓ(x). √ If n is a non-negative integer, Γ(n + 1) = n!. Γ(1/2) = π . ∞ The Method of Frobenious—solution forms: ∞ y1 (x) = xr n=0 an xn (= y2 (x) ?), ∞ 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 Frobenious—useful formula: u′′ + p(x)u′ + q (x)u = Bessel Functions. A. The Bessel equation of order ν : B. Bessel functions: Jν (x) = J−ν (x) = x 2 x 2 ν∞ C ′ [y1 (x) − xp(x)y1 (x) − 2xy1 (x)] . x2 x2 y ′′ + xy ′ + (x2 − ν 2 )y = 0. k =0 −ν ∞ (−1)k k!Γ(ν + k + 1) (−1)k k!Γ(k − ν + 1) x 2 x 2 2k 2k k =0 (cos νπ )Jν (x) − J−ν (x) , if v = 0, 1, 2, . . . . Yν (x) = sin νπ Yn (x) = lim Yν (x), if n = 0, 1, 2, . . . . ν →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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This note was uploaded on 12/15/2009 for the course MATH 527 taught by Professor Staff during the Fall '08 term at Rutgers.

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