Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 3 ppsx

... 12   β α  2 − 1       < 0 for |β/α| < 1 = 0 for |β/α| = 1 > 0 for |β/α| > 1 Thus we see that β α x may be a minimum for |β/α| ≥ 1 and may be a maximum for |β/α| ≤ 1. Jacobi Condition. ... closed curve, and P (t) is a given entire function of t. Exercise 48.41 Solve −  1 0 f(t) t − x dt + −  3 2 f(t) t − x dt = x where this equation is to hold for x in...

Ngày tải lên: 06/08/2014, 01:21

... Figure 3. 13. 2 4 6 8 10 0.2 0.4 0 .6 0.8 1 Figure 3. 13: Plot of x 20 /20!. Note that the error is very small for x < 6, fairly small but non-negligible for x ≈ 7 and large for x > 8. The ... positive for −π < x < 0 and negative for 0 < x < π. Since the sign of y  goes from positive to negative, x = 0 is a relative maxima. See Figure 3. 7. Example 3....

Ngày tải lên: 06/08/2014, 01:21

... dz =  sin 3 z 3  ıπ π = 1 3  sin 3 (ıπ) −sin 3 (π)  = −ı sinh 3 (π) 3 Again the anti-derivative is single-valued. 491 3. e x (sin x cos y cosh y −cos x sin y sinh y) Exercise 9 .3 For an analytic ... max a≤x≤b |f(x)|. 466 with a, b and c complex-valued constants and d a real constant. Substituting z = x + ıy and expanding products yields, a  x 3 + ı3x 2 y − 3...

Ngày tải lên: 06/08/2014, 01:21

... integrand to s ee that there are singularities at the cube roots of 9. z z 3 − 9 = z  z − 3 √ 9  z − 3 √ 9 e ı2π /3  z − 3 √ 9 e −ı2π /3  Let C 1 , C 2 and C 3 be contours around z = 3 √ 9, ... lower and 530 3.  C √ z dz 1. Since sin (cos (z 5 )) is an analytic function inside the unit circle,  C sin  cos  z 5  dz = 0 2. 1 (z 3) (3z−1) has singulari ties at z =...

Ngày tải lên: 06/08/2014, 01:21

... −x)  =  2 π  π 0 x(π −x) sin(nx) dx =  2 π 2 n 3 (1 − (−1) n ) =   2 π 4 n 3 for odd n 0 for even n Thus the expansion is x(π −x) = ∞  n=1 oddn 8 πn 3 sin(nx) for x ∈ [0, π]. In the first graph of Figure ... = 1 √ π x −1 e −x 2 − 1 √ π  ∞ x t −2 e −t 2 dt = 1 √ π x −1 e −x 2 − 1 √ π  − 1 2 t 3 e −t 2  ∞ x + 1 √ π  ∞ x 3 2 t −4 e −t 2 dt = 1 √ π e −x 2  x −1 − 1 2 x...

Ngày tải lên: 06/08/2014, 01:21

... the one, two and three term approximations give relative errors of 0.01, 0.00 06 and 0.000 06, respectively. 167 9 Part V Partial Differential Equations 168 0 Solution 34 .8 For this part we will use ... put Equation 36 . 1 in the form u ξξ = G(ξ, ψ, u, u ξ , u ψ ). ( 36 . 6) We require that the u ξψ and u ψψ terms vanish. That is β = γ = 0 in Equation 36 . 2. This gives us two...

Ngày tải lên: 06/08/2014, 01:21

... = ξ and time t = τ. The disturbance influences the solution for all ξ − ct < x < ξ + ct and t > τ. -1 -0.5 0 0.5 1 x 0 0.2 0.4 0 .6 0.8 1 t 0 0.2 0.4 -1 -0.5 0 0.5 1 x 0 0.2 0.4 0 .6 0.8 1 t Figure ... t Figure 45.1 shows the solution at t = 1/2 and t = 2. -2 -1 1 2 0.1 0.2 0 .3 0.4 0.5 -4 -2 2 4 0.1 0.2 0 .3 0.4 0.5 Figure 45.1: The solution at t = 1/2 and t = 2. Figure...

Ngày tải lên: 06/08/2014, 01:21

... = 3/ 4, t = 7/2 is, u (3/ 4, 7/2) = 1 30 + 7 2 − 3 ∞  n=1 1 n 4 π 4 cos(3nπ/2) cos(7nπ). Note that the summand is nonzero only for even terms. u (3/ 4, 7/2) = 53 15 − 3 16 4 ∞  n=1 1 n 4 cos(3nπ) ... = 53 15 − 3 16 4 ∞  n=1 1 n 4 cos(3nπ) cos(14nπ) = 53 15 − 3 16 4 ∞  n=1 (−1) n n 4 = 53 15 − 3 16 4 −7π 4 720 u (3/ 4, 7/2) = 12727 38 40 2 033 determine the coe...

Ngày tải lên: 06/08/2014, 01:21

... x 1 √ c 2 − 1  The values of the integral for these two values of a are √ 2(x 1 ) 3/ 2 −1 + 3c 2 ± 3c √ c 2 − 1 3( c ± √ c 2 − 1) 3/ 2 . 2095 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 Figure 48.4: sinh(cx) Weierstrass ... yields, x(t) = kt 2 (3T −t) 2 √ 3T 3/ 2 − 1 2 gt 2 . Applying the condition x(T ) = h gives us, k √ 3 T 3/ 2 − 1 2 gT 2 = h, 1 4 g 2 T 4 − k 3 T 3 + ghT 2 + h 2 = 0...

Ngày tải lên: 06/08/2014, 01:21

... few are λ 1 ≈ 0 .67 8298 λ 2 ≈ 7.27 931 λ 3 ≈ 24. 930 2 λ 4 ≈ 54.25 93 λ 5 ≈ 95 .30 57 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 ... equation is φ(x) =        a + bx for λ = 0 a cos  √ λx  + b sin  √ λx  for λ > 0 a cosh  √ −λx  + b sinh  √ −λx  for λ < 0 We see that for λ = 0 and λ &l...

Ngày tải lên: 06/08/2014, 01:21