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formula sheet 1

# formula sheet 1 - solves x a y 00 ax a-1 y bx c y = 0 D...

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Formula Sheet Taylor series (with radii of convergence given): 1 1 - x = 1 + x + x 2 + ··· = X 0 x n , | x | < 1 e x = 1 + x + x 2 2! + x 3 3! + ··· = X 0 x n n ! , | x | < cos x = 1 - x 2 2! + x 4 4! - ··· = X 0 ( - 1) n x 2 n (2 n )! , | x | < sin x = x - x 3 3! + x 5 5! - ··· = X 0 ( - 1) n x 2 n +1 (2 n +1)! , | x | < The Gamma function. For x > 0 , Γ( x ) = Z 0 t x - 1 e - t d t . If x is not 0 or a negative integer, Γ( x +1) = x Γ( x ) . If n is a non-negative integer, Γ( n +1) = n ! . Also Γ(1 / 2) = π . Bessel Functions. A. The Bessel equation of order ν : x 2 y 00 + xy 0 + ( x 2 - ν 2 ) y = 0 . B. The modified Bessel equation of order ν : x 2 y 00 + xy 0 + ( - x 2 - ν 2 ) y = 0 . C. If (i): α = 2 c - a +2 and ν = 1 - a c - a +2 ; and if (ii) u solves the Bessel equation of order ν and b > 0 , then y ( x ) = x ν/α u ± α bx 1 ² solves x a y 00 + ax a - 1 y 0 + bx c y = 0 . If instead u solves the modified Bessel equation of order ν and b < 0 , then y ( x ) = x ν/α u ± α p | b | x 1 ²

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Unformatted text preview: solves x a y 00 + ax a-1 y + bx c y = 0 . D. Bessel functions: J ν ( x ) = ± x 2 ² ν ∞ X n =0 (-1) k k !Γ( ν + k +1) ± x 2 ² 2 k J-ν ( x ) = ± x 2 ²-ν ∞ X n =0 (-1) k k !Γ( k-ν +1) ± x 2 ² 2 k Y ν ( x ) = (cos νπ ) J ν ( x )-J-ν ( x ) sin νπ , if v 6 = 0 , 1 , 2 ,... . Y n ( x ) = lim ν → n Y ν ( x ) , if n = 0 , 1 , 2 ,... . On the exam, the remaining part of this formula sheet will be Appendix C, the Laplace trans-form tables, from the text....
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formula sheet 1 - solves x a y 00 ax a-1 y bx c y = 0 D...

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