Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 2 docx

... + ζ 2 2(2n 2) + ζ 4 2 4·(2n 2) (2n−4) + ··· + ζ n 2 4···n·(2n 2) ···(2n−n)  for even n 2 n−1 n! ζ n+1  1 + ζ 2 2(2n 2) + ζ 4 2 4·(2n 2) (2n−4) + ··· + ζ n−1 2 4···(n−1)·(2n 2) ···(2n−(n−1))  for ... 0.01 65 15 8.71783 · 10 10 8.66 954 · 10 10 0.0 055 25 6 .20 448 · 10 23 6.18384 · 10 23 0.0033 35 2. 9 52 33 · 10 38 2. 9 453 1 · 10 38 0.0 024 45 2. 658 27 ·...

Ngày tải lên: 06/08/2014, 01:21

... + ∞  n=1 nd n R n−1 sin(nθ) 18 15 m=3,n=1 m=3,n =2 m=3,n=3 m =2, n=1 m =2, n =2 m =2, n=3 m=1,n=1 m=1,n =2 m=1,n=3 m=3,n=1 m=3,n =2 m=3,n=3 m =2, n=1 m =2, n =2 m =2, n=3 m=1,n=1 m=1,n =2 m=1,n=3 Figure 37.4: The modes of ... the sum and difference of these solutions to obtain φ = exp   α 2 − β 2  t ± α a x  cos sin  2 βt ± β a x  1817 m =2, n=1 m =2, n =2 m =2, n=3 m=1,...

Ngày tải lên: 06/08/2014, 01:21

... = 2x 2 x 2 + βx + χ Now we use the fact that f(x) is even to conclude that q(x) is even and thus β = 0. f(x) = 2x 2 x 2 + χ Finally, we use that f(1) = 1 to determine χ. f(x) = 2x 2 x 2 + 1 20 Einstein ... there and that χ = 0. f(x) = ax 2 x 2 + βx + χ We n ote that f (x) → 2 as x → ∞. This d etermine s the parameter a. lim x→∞ f(x) = lim x→∞ ax 2 x 2 + βx + χ = l...

Ngày tải lên: 06/08/2014, 01:21

... A = 0, 2 ln t + 1 2 t 2 + c, A = 0 − 1 2 t 2 + ln t + c, A = 2 y =      1 A + t 2 A +2 + ct −A , A = 2 ln t + 1 2 t 2 + c, A = 0 − 1 2 + t 2 ln t + ct 2 , A = 2 For positive A, the ... are: y =      1 A + t 2 A +2 , A > 0 1 A + t 2 A +2 + ct −A , A < 0, A = 2 − 1 2 + t 2 ln t + ct 2 , A = 2 Equations in the Complex Plane Solution 14....

Ngày tải lên: 06/08/2014, 01:21

... − 1− √ 5 2 1+ √ 5 2 √ 5 = 1+ √ 5 2 1+ √ 5 2 √ 5 = 1 √ 5 Substitute this result into the equation for c 2 . c 2 = 1 r 2  1 − 1 √ 5 r 1  = 2 1 − √ 5  1 − 1 √ 5 1 + √ 5 2  = − 2 1 − √ 5  1 − √ 5 2 √ 5  = − 1 √ 5 Thus ... solution b n = n  j=0  − 4j 2 − 2j + 1 (2j + 2) (2j + 1)  . Thus we have that a n =   n /2 j=0  − 4j 2 −2...

Ngày tải lên: 06/08/2014, 01:21

... cases for even and odd n separately. a 2n = − a 2n 2 2n = a 2n−4 (2n)(2n − 2) = (−1) n a 0 (2n)(2n − 2) ···4 · 2 = (−1) n a 0  n m=1 2m , n ≥ 0 a 2n+1 = − a 2n−1 2n + 1 = a 2n−3 (2n + 1)(2n − 1) = ... =  x 3 3  1 −1 = 2 3  1 −1 P 2 (x)P 2 (x) dx =  1 −1 1 4  9x 4 − 6x 2 + 1  dx =  1 4  9x 5 5 − 2x 3 + x  1 −1 = 2 5  1 −1 P 3 (x)P 3 (x) dx =  1 −1 1 4  25...

Ngày tải lên: 06/08/2014, 01:21

... v]   x=a 27 .2 Formally Self-Adjoint Operators Example 27 .2. 1 The linear operator L[y] = x 2 y  + 2xy  + 3y has the adjoint operator L ∗ [y] = d 2 dx 2 (x 2 y) − d dx (2xy) + 3y = x 2 y  + ... 4xy  + 2y −2xy  − 2y + 3y = x 2 y  + 2xy  + 3y. In Example 27 .2. 1, the adjoint operator is the same as the operator. If L = L ∗ , the operator is said to be formally self...

Ngày tải lên: 06/08/2014, 01:21

... that ∞  n=1 n  a 2 n + b 2 n + c 2 n + d 2 n  = ∞  n=1 n 2  a 2 n + b 2 n + c 2 n + d 2 n  14 15 -1 -0 .5 0 .5 1 -0 .2 -0.1 0.1 0 .2 -1 -0 .5 0 .5 1 -0 .2 -0.1 0.1 0 .2 Figure 28 .13: The odd and even periodic ... inequality |ab| ≤ 1 2 |a 2 + b 2 |, which follows from expanding (a − b) 2 ≥ 0. A ≤ π 2 ∞  n=1 n  a 2 n + b 2 n + c 2 n + d 2...

Ngày tải lên: 06/08/2014, 01:21

... transform of ˆy 1 . y 1 = t 3 6 − t 2 2 + 1 2 e −t + 1 2 sin t − 1 2 cos t. We can find y 2 and y 3 by differentiating the expression for y 1 . y 2 = t 2 2 − t − 1 2 e −t + 1 2 cos t + 1 2 sin ... − 1 2  ∞ 0 e −rt √ π −ı √ r e 2 √ a √ r (−dr) = 1 2 √ π  ∞ 0 e −rt 1 √ r  e − 2 √ a √ r + e 2 √ a √ r  dr = 1 2 √ π  ∞ 0 1 √ r e −rt 2 cos  2 √ a √ r  dr...

Ngày tải lên: 06/08/2014, 01:21

... convolution of u(x) and e −ax 2 . Hint 32. 12 Hint 32. 13 Hint 32. 14 Hint 32. 15 Hint 32. 16 Hint 32. 17 Hint 32. 18 Hint 32. 19 Hint 32. 20 157 9 Solution 32. 6 F −1 s [ ˆ f s (ω)G c (ω)] = 2  ∞ 0 ˆ f s (ω)G c (ω) ... transform of the differential equ ation, solve for ˆ G and then invert. G  + 2 G  +  β 2 + µ 2  G = δ(x − ξ) −ω 2 ˆ G + 2 ω ˆ G +  β 2 + µ 2  ˆ...

Ngày tải lên: 06/08/2014, 01:21