{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

527formf - 642:527 FORMULA SHEET FOR FINAL EXAM FALL 2009...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 642:527 FORMULA SHEET FOR FINAL EXAM FALL 2009 Taylor series (with radii of convergence given): 1 1 − x = 1 + x + x 2 + ··· = ∞ summationdisplay x n , | x | < 1 e x = 1 + x + x 2 2! + x 3 3! + ··· = ∞ summationdisplay x n n ! , | x | < ∞ cos x = 1 − x 2 2! + x 4 4! − ··· = ∞ summationdisplay ( − 1) n x 2 n (2 n )! , | x | < ∞ sin x = x − x 3 3! + x 5 5! − ··· = ∞ summationdisplay ( − 1) n x 2 n +1 (2 n + 1)! , | x | < ∞ The Gamma function. For x > 0, Γ( x ) = integraltext ∞ 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 !. Γ(1 / 2) = √ π . The Method of Frobenious—solution forms: y 1 ( x ) = x r ∞ summationdisplay n =0 a n x n (= y 2 ( x ) ?), y 2 ( x ) = y 1 ( x )(ln x ) + x r 1 ∞ summationdisplay n =1 b n x n , y 2 ( x ) = Cy 1 ( x )(ln x ) + x r 2 ∞ summationdisplay n =0 b n x n ....
View Full Document

{[ snackBarMessage ]}