527form2

# 527form2 - 642:527 FORMULA SHEET FOR EXAM 2 FALL 2009...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 642:527 FORMULA SHEET FOR EXAM 2 FALL 2009 Various Fourier-type expansions: We write f (x) ∼ series to indicate that the given series is some Fourier-type expansion of f (x). All the formulas for coeﬃcients come from the formula f , ϕn cn = ϕn , ϕn with ϕ1 , ϕ2 , . . . a complete orthogonal set for Cp [−ℓ, ℓ] or Cp [0, L]. ∞ nπx nπx an cos f (x) ∼ a0 + ; + bn sin ℓ ℓ n=1 (1) ℓ 1ℓ 1ℓ nπx nπx 1 dx, bn = dx. f (x) dx, an = f (x) cos f (x) sin a0 = 2ℓ −ℓ ℓ −ℓ ℓ ℓ −ℓ ℓ ∞ f (x) ∼ n=−∞ cn einπx/ℓ ; ∞ cn = 1 2ℓ ℓ f (x)e−inπx/ℓ dx. −ℓ (2) nπx f (x) ∼ a0 + an cos L n=1 1 a0 = L ∞ L (0 < x < L); nπx dx. f (x) cos L bn = 2 L L f (x) dx, 0 2 an = L L 0 nπx dx. L nπx dx. 2L nπx dx. 2L (3) f (x) ∼ n=1 bn sin nπx L nπx 2L nπx 2L (0 < x < L); f (x) sin 0 L (4) f (x) ∼ n=1,3,5,... an cos (0 < x < L); an = 2 L 2 L f (x) cos 0 L (5) f (x) ∼ n=1,3,5,... bn sin (0 < x < L); bn = f (x) sin 0 (6) Some antiderivatives: e−ax (a sin(bx) + b cos(bx)) + C a2 + b2 e−ax e−ax cos(bx) dx = 2 (−a cos(bx) + b sin(bx)) + C a + b2 sin(bx) x cos(bx) cos(bx) x sin(bx) + +C x sin(bx) dx = − +C x cos(bx) dx = 2 b b b2 b Some trig identities: e−ax sin(bx) dx = − cos2 x = 1 + cos(2x) 1 − cos(2x) sin2 x = 2 2 cos(A) cos(B ) = (1/2) [cos(A + B ) + cos(A − B )] sin(A) cos(B ) = (1/2) [sin(A + B ) + sin(A − B )] sin(A) sin(B ) = (1/2) [cos(A − B ) − cos(A + B )] Sturm-Liouville problem: L [p(x)y ′ ]′ + q (x)y + λw(x)y = 0, f (x), y (x) w = 0 f (x)g(x)w(x) dx ...
View Full Document

## This note was uploaded on 12/15/2009 for the course MATH 527 taught by Professor Staff during the Fall '08 term at Rutgers.

Ask a homework question - tutors are online