Exam1Equations

# Exam1Equations - , ≤ x ≤ 1 Legendre Equation: d dx ±...

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Exam #1 Equations Trigonometric Identities: sin( x ± y ) = sin x cos y ± cos x sin y cos( x ± y ) = cos x cos y sin x sin y tan( x ± y ) = tan x ± tan y 1 tan x tan y sin 2 x + cos 2 x = 1 , cosh 2 x - sinh 2 x = 1 cos 2 x = cos 2 x - sin 2 x, sin 2 x = 2 sin x cos x cos 2 x = 1 2 (1 + cos 2 x ) , sin 2 x = 1 2 (1 - cos 2 x ) cos 3 x = 3 4 cos x + 1 4 cos(3 x ) , sin 3 x = 3 4 sin x - 1 4 sin(3 x ) e ix = cos x + i sin x sin x = e ix - e - ix 2 i , cos x = e ix + e - ix 2 sinh x = e x - e - x 2 , cosh x = e x + e - x 2 sin( iz ) = i sinh z, cos( iz ) = cosh z sinh( iz ) = i sin z, cosh( iz ) = cos z Integrals: Z x sin( ax ) dx = sin( ax ) a 2 - x cos( ax ) a + C Z x cos( ax ) dx = cos( ax ) a 2 + x sin( ax ) a + C Sturm-Liouville Equation: d dx ± p ( x ) du dx ² + [ q ( x ) + λr ( x )] u = 0 a 0 ( x ) d 2 u dx 2 + a 1 ( x ) du dx + [ a 2 ( x ) + λa 3 ( x )] u = 0 p ( x ) = exp ±Z a 1 ( x ) a 0 ( x ) dx ² ,q ( x ) = a 2 ( x ) a 0 ( x ) p ( x ) ,r ( x ) = a 3 ( x ) a 0 ( x ) p ( x ) Fourier Series: f ( x ) = a 0 2 π + X n =1 a n cos( nx ) π + X m =1 b m sin( mx ) π a 0 = ³ f ( x ) , 1 2 π ´ = 1 2 π Z 2 π 0 f ( x ) dx a n = ³ f ( x ) , cos( nx ) π ´ = 1 π Z 2 π 0 f ( x ) cos( nx ) dx b m = ³ f ( x ) , sin( mx ) π ´ = 1 π Z 2 π 0 f ( x ) sin( mx ) dx Bessel Equation: d dx ± x du dx ² + ± - ν 2 x + μ 2 x ² u = 0
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Unformatted text preview: , ≤ x ≤ 1 Legendre Equation: d dx ± (1-x 2 ) du dx ² + ν ( ν + 1) u = 0 ,-1 ≤ x ≤ 1 P ( x ) = 1 ,P 1 ( x ) = x,P 2 ( x ) = 1 2 (3 x 2-1) ,... ( ν + 1) P ν +1 ( x )-(2 ν + 1) xP ν ( x ) + νP ν-1 ( x ) = 0 Chebyshev Equation: d dx ± (1-x 2 ) 1 / 2 du dx ² + ν 2 (1-x 2 )-1 / 2 u = 0 ,-1 ≤ x ≤ 1 T ( x ) = 1 ,T 1 ( x ) = x,T 2 ( x ) = 2 x 2-1 ,... T ν +1 ( x )-2 xT ν ( x ) + T ν-1 ( x ) = 0 1...
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## This note was uploaded on 09/24/2010 for the course MMAE 501 taught by Professor Kevincassel during the Spring '10 term at Illinois Tech.

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