Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4 docx

... of 4 3 √ π cm and a height of 4 3 √ π . 106 a x x x x x x∆ 1 2 3 i n-2 n-1 b f( ) ξ 1 Figure 4. 2: Divide -and- Conquer Strategy for Approximating a Definite Integral. 4. 2.2 Properties Linearity and ... of x and an increasing function of δ for positive x and δ. Thus for any fixed δ, the maximum value of √ x + δ − √ x is bounded by √ δ. Therefore on the interval (0, 1), a su...

Ngày tải lên: 06/08/2014, 01:21

... sinh(1) 4. We consider the integral  C e zt z 2 (z + 1) dz. There are singularities at z = 0 and z = −1. Let C 1 and C 2 be contours around z = 0 and z = −1. See Figure 11.6. We deform C onto C 1 and ... analytic in the domain D : |z| < R that satisfy f(0) = e ı and |f(z)| ≤ 1 for all z in D. Exercise 11.13 Let f(z) =  ∞ k=0 k 4  z 4  k and evaluate the following con...

Ngày tải lên: 06/08/2014, 01:21

... R(x)y = 0. (16 .4) An exact equation can be written in the form: d dx [a(x)y  + b(x)y] = 0. If Equation 16 .4 is exact, then we can write it in the form: d dx [P (x)y  + f(x)y] = 0 for some function ... differentiation to write the equation in standard form: P (x)y  + (P  (x) + f(x)) y  + f  (x)y = 0 (16.5) We equate the coefficients of Equations 16 .4 and 16.5 to obtain a set of e...

Ngày tải lên: 06/08/2014, 01:21

... A stands for either I or K. 1 649 1 2 3 4 2 4 6 8 10 Figure 34. 4: Modified Bessel Functions I ν and K ν are linearly independent for all ν. In Figure 34. 4 I 0 and K 0 are plotted in solid and dashed ... + 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)·...

Ngày tải lên: 06/08/2014, 01:21

... we determine x and y in terms of ξ and ψ. x = −  3 4 (ξ + ψ)  1/3 , y = ψ −ξ 2 We calculate the derivatives of ξ and ψ. ξ x = √ −x =  3 4 (ξ + ψ)  1/6 , ξ y = −1 ψ x =  3 4 (ξ + ψ)  1/6 , ... = 0 for 0 < x < l, with boundary conditions φ(0, t) = 0 for t > 0, and φ(l, t) + φ x (l, t) = 1 for t > 0. Obtain two series solutions for this problem, one which...

Ngày tải lên: 06/08/2014, 01:21

... function at the point x = p is G(p|ξ) = − 1 4 |p − ξ| + const. 1 849 40 .4 Hints Hint 40 .1 Hint 40 .2 Hint 40 .3 Hint 40 .4 Hint 40 .5 Hint 40 .6 Hint 40 .7 Hint 40 .8 1 846 ... y) = 0 and u(x, 0) = 0, u(x, 1) =  sin(nπx). Is this well posed? Exercise 40 .4 Use the fundamental solutions for the Laplace equation ∇ 2 G = δ(x − ξ) 1 843 We seek a solution of the form, u(x,...

Ngày tải lên: 06/08/2014, 01:21

... off the pin. For L = 2, c 1 has the values 0 .42 65 and 0.75 24. The larger value of c 1 gives the smaller potential energy. The position of the end of the rope is z = −0.9 248 . Solution 48 .9 Using ... approximation and the exact solution are plotted in Figure 48 .5. The value of the functional is J = − 5 216 ≈ −0.023 148 1. 0.2 0 .4 0.6 0.8 1 0.01 0.02 0.03 0. 04 0.05 0.06 0.07 Fi...

Ngày tải lên: 06/08/2014, 01:21

... of, 1 349 Fourier convolution theorem, 15 54 Fourier cosine series, 1 344 Fourier cosine transform, 1562 of derivatives, 15 64 table of, 22 54 Fourier series, 1330 and Fourier transform, 1539 uniform ... 1 345 Fourier sine series, 142 9 Fourier sine transform, 1563 of derivatives, 15 64 table of, 2255 Fourier transform alternate definitions, 1 544 closure relation, 1552 convolution t...

Ngày tải lên: 06/08/2014, 01:21

... . . 1910 43 .10Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1911 44 Transform Methods 1918 44 .1 Fourier Transform for Partial Differential ... also contains the word Scientists or Engineers ” the advanced book may be quite suitable for actually learning the material. xxvii 43 .3 The Method of Characteristics and t...

Ngày tải lên: 06/08/2014, 01:21

... a to b and n is a unit vector that is orthogonal to a and b and in the direction s uch that the ordered triple of vectors a, b and n form a right-handed system. 29 a b b θ b Figure 2. 14: The ... a i b i Solution 2.3 Since a · b = |a||b|cos θ, we have θ = arccos  a · b |a||b|  40 Part II Calculus 47 2 .4 Hints The Dot and Cross Product Hint 2.1 First prove the distributive l...

Ngày tải lên: 06/08/2014, 01:21