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

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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 ...
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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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