Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 7 docx

... 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 geometry in which A is a function of ... dξ =  x a h x (x, ξ) dξ + h(x, x). Hint 39.6 1834 Solution 37. 37 We will assume that both α and β are nonzero. The cases of real and pure imaginary have already been cove...

Ngày tải lên: 06/08/2014, 01:21

... 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 extension of x(1 − x), 0 ≤ x ≤ 1. The ... the form p 2 y  + p 1 y  + p 0 y = µy, for a ≤ x ≤ b, α 1 y(a) + α 2 y  (a) = 0, β 1 y(b) + β 2 y  (b) = 0, where the p j are real and continuous and p 2 > 0 on [...

Ngày tải lên: 06/08/2014, 01:21

... error 5 24 23.6038 0.01 65 15 8 .71 783 · 10 10 8.66 954 · 10 10 0.0 055 25 6.20448 · 10 23 6.18384 · 10 23 0.0033 35 2. 952 33 · 10 38 2.9 453 1 · 10 38 0.0024 45 2. 658 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 0.1 0.2 0.3 Figure...

Ngày tải lên: 06/08/2014, 01:21

... This is zero if and only if u 0 = u 1 = u 2 = u 3 = 0. Thus there 2 17 -10 -5 5 10 - 15 -10 -5 5 Figure 6.23: (z) − (z) = 5 Solution 6.16 | e ıθ −1| = 2  e ıθ −1  e −ıθ −1  = 4 1 − e ıθ − e −ıθ +1 ... Figure 7. 10. The modulus and principal argument of f(z) = z are plotted in Figure 7. 11. Example 7 .5. 2 Consider the function f(z) = z 2 . In Cartesian co ordinates and...

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... + 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 the integral.  C cot z coth z z 3 dz = −ı 14 45 π 13.2 Cauchy ... union of C p and C  . (C p stands for Principal value Contour; C i stands for Indented Contour.) C i is an indented contour that avoids the first order pole at z = 1. Figur...

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... Equations 10 17 19.6 Hints The Constant Coefficient Equation Normal Form Hint 19.1 Transform the equation to normal form. Transformations of the Independent Variable Integral Equations Hint 19.2 Transform ... that satisfy the left and right boundary conditions are c 1 (x − a) and c 2 (x − b). Thus the Green’s function has the form G(x|ξ) =  c 1 (x − a), for x ≤ ξ c 2 (x − b), for x ≥...

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... difference equation for b n that is of order one less than the equation for a n . 1 178 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 r 1  = 2 1 − √ 5  1 − 1 √ 5 1 + √ 5 2  = − 2 1 − √ 5  1 − √ 5 2 √ 5  = − 1 √ 5 Thus the n th term in the Fibonacci sequence has the formu...

Ngày tải lên: 06/08/2014, 01:21

... eigenfunction φ. Green’s formula states φ|L[φ] − L[φ]|φ = 0 φ|λφ − λφ|φ = 0 (λ − λ)φ|φ = 0 Since φ ≡ 0, φ|φ > 0. Thus λ = λ and λ is real. 1319 27. 7 Hints Hint 27. 1 13 27 2. For k = n, φ k , ... the formula to obtain information about the eigenvalues before we solve a problem. Example 27. 4.2 Consider the self-adjoint eigenvalue problem −y  = λy, y(0) = y(π) = 0. 1...

Ngày tải lên: 06/08/2014, 01:21

... ıa  ∞ 0 , for (s) > 0 = 1 ı2  1 s − ıa − 1 s + ıa  L[sin(at)] = a s 2 + a 2 for (s) > 0 150 7 Since ˆ f  (s) is defined for (s) > 0, ˆ f(s) is analytic for (s) > 0. Let σ be real and ... have 1 ı2π   B +  L + +  C  +  L −  e st  π s  1/2 e −2(as) 1/2 ds = 0. 152 7 By inspection we see that this is satisfied for f(t) = 1 for 0 < t < T . We c...

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... b. 159 7 Hint 32.10 Hint 32.11 The left side is the 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 1 57 9 Solution ... f(−x)]) where F, F c and F s are respectively the Fourier transform, Fourier cosine transform and Fourier sine transform. Hint, Solution 1 57 6 ... Since 159 5 Exercise 32.2...

Ngày tải lên: 06/08/2014, 01:21