Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 5 pdf

... -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -0 .4 -0.2 0.2 0 .4 Figure 28.7: Three Term Approximation for a Function with Jump Discontinuities and a Continuous Function. A ... + 2  1 1/2 (1 − x) sin(nπx) dx = 4 (nπ) 2 sin(nπ/2) =  4 (nπ) 2 (−1) (n−1)/2 for odd n 0 for even n. 1 356 -3 -2 -1 1 2 3 -1 .5 -1 -0 .5 0 .5 1 1 .5 Figure 28 .5: Fourier Sine Series. The...

Ngày tải lên: 06/08/2014, 01:21

... + √ 87  1/3  2   ≈ (0 .58 9 755 , 0, 0. 347 81). The closest point is shown graphically in Figure 5. 10. -1 -0 .5 0 0 .5 1 -1 -0 .5 0 0 .5 1 0 0 .5 1 1 .5 2 -1 -0 .5 0 0 .5 1 0 0 .5 1 1 .5 2 Figure 5. 10: Paraboloid, ... 2     + C 147 Solution 4. 4  cos x sin x dx =  1 sin x d(sin x) dx dx = ln |sin x| Solution 4. 5  x 2 x 3 − 5 dx =  1 x 3 − 5 1 3 d(x 3 ) dx...

Ngày tải lên: 06/08/2014, 01:21

... problem y  + 5y  + 6y = 0; y(0) = 1, y  (0) = V. For what values of V does y(t) remain nonnegative for all t > 0? 952 3 4 5 6 -10 10 20 30 40 50 Figure 17.2: The solution of y  − 2y  + 5y = 0, ... a ln x ξ  ,  x ξ  −a ξ ln x ξ  if a 2 = b. 943 1 2 3 4 5 -30 - 25 -20 - 15 -10 -5 Figure 17.1: The solution of 6y  − 5y  + y = 0, y(0) = 4, y  (0) = 0. The...

Ngày tải lên: 06/08/2014, 01:21

... continuous. -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 -1 -0 .5 0 .5 1 Figure 25. 1: Polynomial Approximations to cos(πx). 1286 0 1 2 3 4 5 0. 25 0 .5 0. 75 1 1. 25 1 .5 1. 75 2 Figure 24. 1: Plot of K 0 (x) and it’s ... 21) π 4 P 4 (x) = 1 05 8π 4 [(3 15 − 30π 2 )x 4 + ( 24 2 − 270)x 2 + (27 − 2π 2 )] The cosine and this polynomial are plotted in the sec...

Ngày tải lên: 06/08/2014, 01:21

... are λ 1 ≈ 0.678298 λ 2 ≈ 7.27931 λ 3 ≈ 24. 9302 λ 4 ≈ 54 . 259 3 λ 5 ≈ 95. 3 057 The eigenfunctions are, φ n (x) = v  (1; λ n )u(x; λ n ) − u  (1; λ n )v(x; λ n ). Solution 48 .32 1. First note that sin(kx) ... − 4 λ ˆ f(2x) + 4 λ ˆ φ(2x) (48 .9) We eliminate ˆ φ between (48 .8) and (48 .9).  1 − πλ 2 4  φ(x) = f(x) + λπ ˆ f(2x) φ(x) = f(x) + λ  ∞ 0 f(s) cos(2xs) ds 1 − πλ...

Ngày tải lên: 06/08/2014, 01:21

... . . 1910 43 .10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1911 44 Transform Methods 1918 44 .1 Fourier Transform for Partial Differential ... . . . . . . . . . . . . . 155 7 32 .4. 5 Shift Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 9 32 .4. 6 Fourier Transform of...

Ngày tải lên: 06/08/2014, 01:21

... , converges for α > 1 and diverges for α ≤ 1. Hint, Solution 5 64 Example 12.3.2 Convergence and Uniform Convergence. Consider the series log(1 − z) = − ∞  n=1 z n n . This series converges for |z| ... large. Thus this series is not uniformly convergent in the domain |z| ≤ 1, z = 1. The series is uniformly convergent for |z| ≤ r < 1. 54 5 12.2.2 Uniform Convergence a...

Ngày tải lên: 06/08/2014, 01:21

... − z 4 6 + ··· 1 − z 4 90 + ··· = 1 z 5  1 − z 4 6 + ···  1 + z 4 90 + ···  = 1 z 5  1 − 7 45 z 4 + ···  = 1 z 5 − 7 45 1 z + ··· Thus we see that the residue is − 7 45 . N ow we can evaluate ... − z 2 2 + z 4 24 − ···  1 + z 2 2 + z 4 24 + ···  z 3  z − z 3 6 + z 5 120 − ···  z + z 3 6 + z 5 120 + ···  = 1 − z 4 6 + ··· z 3  z 2 + z 6  −1 36...

Ngày tải lên: 06/08/2014, 01:21

... difference equation for b n that is of order one less than the equation for a n . 1178 1 2 3 4 5 6 -1 -0 .5 0 .5 1 1 .5 1 2 3 4 5 6 -1 -0 .5 0 .5 1 1 .5 Figure 23.3: The graph of approximations and numerical ... − 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 − √...

Ngày tải lên: 06/08/2014, 01:21

... =  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  25x 6 − 30x 4 + 9x 2  dx =  1 4  25x 7 7 − 6x 5 + 3x 3  1 −1 = 2 7 Solution ... singular point at ξ = 0, the equation for y(x) has an irregular singul ar point at infinity. 1 242 Exercise 23 .5 Find the...

Ngày tải lên: 06/08/2014, 01:21