Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 8 doc

... -5 0 5 x 0 2 4 6 t -1 -0 .5 0 0 .5 1 u -5 0 5 x Figure 41.4: Solution of the wave equation. We integrate the Heaviside function.  b a H(x −c) dx =      0 for b < c b −a for a > ... −1)πy 2b  dy With u and v determined, the solution of the original problem is w = u + v. 1 85 8 Direct Solution. D’Alembert’s solution is valid for all x and t. We formally substitu...

Ngày tải lên: 06/08/2014, 01:21

... error 5 24 23.60 38 0.01 65 15 8. 71 783 · 10 10 8. 66 954 · 10 10 0.0 055 25 6.204 48 · 10 23 6. 183 84 · 10 23 0.0033 35 2. 952 33 · 10 38 2.9 453 1 · 10 38 0.0024 45 2. 6 58 27 · 10 54 2. 653 35 · 10 54 0.0019 In ... J −n (z) and J n (z) are not linearly independent for integer n. 16 25 2 4 6 8 10 12 14 -0.4 -0.2 0.2 0.4 0.6 0 .8 1 5 10 15 20 -0.2 -0.1 0...

Ngày tải lên: 06/08/2014, 01:21

... symmetric with respect to x and ξ. We add the constraint that the Green 1 85 1 38. 2 Hints Hint 38. 1 Hint 38. 2 182 6 We find homogeneous solutions which respectively satisfy the left and right homogeneous ... a 2 φ xx , a 2 = k cρ , (39.2) so that it is valid for diffusion in a non-homogeneous medium for which c and k are functions of x and φ and so that it is valid for a g...

Ngày tải lên: 06/08/2014, 01:21

... of, 52 5 series, 52 5 comparison test, 52 9 convergence of, 52 5, 52 6 Gauss’ test, 53 6 geometric, 52 7 integral test, 53 0 Raabe’s test, 53 5 ratio test, 53 1 residuals, 52 7 root test, 53 3 tail of, 52 6 set, ... transform alternate definitions, 154 4 closure relation, 155 2 convolution theorem, 155 4 of a derivative, 155 3 Parseval’s theorem, 155 7 shift property, 155 9 table...

Ngày tải lên: 06/08/2014, 01:21

... are an infinite set of rational numbers for which ı z has 1 as one of its values. For example, ı 4 /5 = 1 1 /5 =  1, e ı2π /5 , e ı4π /5 , e ı6π /5 , e 8 /5  7 .8 Riemann Surfaces Consider the mapping ... See Figure 7. 18 and Figure 7.19 for plots of the real and imaginary parts of the cosine and sine, respectively. Figure 7.20 shows the modulus of the cosine and the s...

Ngày tải lên: 06/08/2014, 01:21

... -1 -0 .5 0 .5 1 0 .5 1 1 .5 2 2 .5 3 Figure 7 .56 : The principal branch of the arc cosine, Arccos(x). Solution 7.33 Arccos(x) is shown in Figure 7 .56 for real variables in the range ... z 1/2 and −1 on the other branch.) For this branch we introduce a branch cut connecting z = 1 with the point at infinity. (See Figure 7 . 58 .) 1 =1 1 =-1 1/2 1/2 Figure 7 . 58 : Branch cuts for...

Ngày tải lên: 06/08/2014, 01:21

... convergent. 17. ∞  n=1 n 8 + 4n 4 + 8 3n 9 − n 5 + 9n = ∞  n=1 1 n 1 + 4n −4 + 8n 8 3 − n −4 + 9n 8 > 1 4 ∞  n=1 1 n We s ee that the series is divergent by comparison to the harmonic series. 18. ∞  n=1  1 n − 1 n ... comparison test. 58 6 17. ∞  n=1 n 8 + 4n 4 + 8 3n 9 − n 5 + 9n Use the comparison test. 18. ∞  n=1  1 n − 1 n + 1  Use the comparison test. 19...

Ngày tải lên: 06/08/2014, 01:21

... and expand in partial fractions. Hint 13.6 Hint 13.7 Cauchy Principal Value for Real Integrals Hint 13 .8 680 f(a) = 2π √ 1 − a 2 Complex-Valued a. We note that the integral converges except for ... order 686 Hint, Solution Exercise 13.23 By methods of contour integration find  ∞ 0 dx x 2 + 5x + 6 [ Recall the trick of considering  Γ f(z) log z dz with a suitably chosen contour Γ...

Ngày tải lên: 06/08/2014, 01:21

... 0, α = 1. 1072 0 .5 1 -0.3 -0.2 -0.1 0.1 0 .5 1 -0.3 -0.2 -0.1 0.1 0 .5 1 -0.3 -0.2 -0.1 0.1 0 .5 1 -0.3 -0.2 -0.1 0.1 Figure 21.3: Plot of G(x|0. 05) ,G(x|0. 25) ,G(x|0 .5) and G(x|0. 75) . Thus the Green ... 1, . . . , n, and W [y 1 , y 2 , . . . , y n ](x) is the Wronskian of {y 1 (x), . . . , y n (x)}. 1070 -4 -2 2 4 0. 05 0.1 0. 15 0.2 0. 25 0.3 -4 -2 2 4 -0.3 -0. 25 -0.2 -0...

Ngày tải lên: 06/08/2014, 01:21

... the problem for u, (and hence the problem for y). As a check, then general solution for y is y = − 1 3 cos 2x + c 1 cos x + c 2 sin x. 11 15 We guess a particular solution of the form y p = t e −t (a ... t, where c 1 and c 2 are arbitrary constants and a and b are any conveniently chosen points. 2. Using the result of part (a) show that the solution satisfying the initial cond...

Ngày tải lên: 06/08/2014, 01:21