527form1

527form1 - 642:527 FORMULA SHEET FOR EXAM 1 FALL 2007...

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Unformatted text preview: 642:527 FORMULA SHEET FOR EXAM 1 FALL 2007 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! x4 x2 + − ··· = cos x = 1 − 2! 4! sin x = x − 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) = x r n=0 an xn , ∞ y2 (x) = y1 (x)(ln x) + xr1 n=1 bn xn , y2 (x) = Cy1 (x)(ln x) + xr2 n=0 bn xn . Bessel Functions. A. The Bessel equation of order ν : x2 y ′′ + xy ′ + (x2 − ν 2 )y = 0. B. If u solves the Bessel equation of order ν , and b > 0, then √ a solves y ′′ + y ′ + bxc−a y = 0, y (x) = xν/α u α bx1/α x where α= C. Bessel functions: x Jν (x) = 2 x J−ν (x) = 2 Yν (x) = ν 2 c−a+2 ∞ and ν= 1−a . c−a+2 2k (−1)k k!Γ(ν + k + 1) n=0 (−1)k k!Γ(k − ν + 1) n=0 ∞ x 2 x 2 −ν 2k (cos νπ )Jν (x) − J−ν (x) , if v = 0, 1, 2, . . . . 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 09/29/2009 for the course 642 527 taught by Professor Speer during the Fall '07 term at Rutgers.

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