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

... ellipse. 187 6. 8 Hints Complex Numbers Hint 6. 1 Hint 6. 2 Hint 6. 3 Hint 6. 4 Hint 6. 5 Hint 6. 6 Hint 6. 7 The Complex Plane Hint 6. 8 Hint 6. 9 208 Part III Functions of a Complex Variable 179 6. 7 Exercises Complex ... and division. Example 6. 3.1 We write 5 + ı7 in polar form. 5 + ı7 = √ 74 e ı arctan(5,7) We write 2 e ıπ /6 in Cartesian form. 2 e ıπ /6 = 2 cos  π 6...

Ngày tải lên: 06/08/2014, 01:21

... 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 C 2 .  C =  C 1 +  C 2 520 11.4 Exercises Exercise ... converges if and only i f for any  > 0 there exists an N such that |a n − a m | <  for all n, m > N. The Cauchy convergence criterion is...

Ngày tải lên: 06/08/2014, 01:21

... convergence with the Cauchy-Hadamard formula. R = 1 lim sup k  |k + 2 k | = 1 lim sup 2 k  |1 + k/2 k | = 1 2 The series converges for |z| < 1/2. 60 0 16. ∞  n=1 (n!) 2 (n 2 )! We app ly ... order of the polynomial. 2 6 12 20 4 6 8 2 2 We s ee that the polynomial is second order. p(n) = an 2 + bn + c. We solve for the coefficients. a + b + c = 2 4a + 2b + c = 6 9a + 3b + c =...

Ngày tải lên: 06/08/2014, 01:21

... x 2 (x −4 u(x)) 2 = 0 x −4 u  − 8x −5 u  + 20x 6 u + x 6 u 2 = 0 x 2 u  − 8xu  + 20u + u 2 = 0. We see that the equation for u is equidimensional-in-x. 1000 For a 2 > b then the general solution ... b. Solution 17.15 For a = 0, two linearly independent solutions of y  − a 2 y = 0 are y 1 = e ax , y 2 = e −ax . For a = 0, we have y 1 = e 0x = 1, y 2 = x e 0x = x. 9 76...

Ngày tải lên: 06/08/2014, 01:21

... the form of a particular solution. This form will contain some unknown parameters. We substitute this form into the differential equation to determine the parameters and thus determine a particular ... are y 1 = e x , and y 2 = e −x . The Green function has the form G(x|ξ) =  c 1 e x +c 2 e −x for x < ξ d 1 e x +d 2 e −x for x > ξ. Since the solution must be bounded for all x,...

Ngày tải lên: 06/08/2014, 01:21

... no value of a for which both cos a and sin a vanish, the system is not orthogonal for any interval of length π. 2. First note that  π 0 cos nx dx = 0 for n ∈ N. If n = m, n ≥ 1 and m ≥ 0 then  π 0 cos ... dx = 2 πn sin(nπ/2) =  2 πn (−1) (n−1)/2 , for odd n 0 for even n The Fourier cosine series of f(x) is f(x) ∼ 1 2 + 2 π ∞  n=0 (−1) n 2n + 1 cos((2n + 1)x). 2. The N th pa...

Ngày tải lên: 06/08/2014, 01:21

... + 4 3 P 2 (cos θ) exp  − 6a 2 R 2 t  Ψ(θ, t) = − 1 3 +  2 cos 2 θ − 2 3  exp  − 6a 2 R 2 t  Solution 37.34 Since we have homogeneous boundary conditions at x = 0 and x = 1, we will expand the solution ... − Y  Y = −λ We have differential equations for X and Y . X  + λX = 0, X(0) = X(1) = 0 Y  − λY = 0, Y (0) = 0 The eigenvalues and orthonormal eigenfunctions for X are λ n...

Ngày tải lên: 06/08/2014, 01:21

... . 1 566 32.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 1 568 32.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms ... . . . . . . . . . 1 560 32 .6 The Fourier Cosine and Sine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 562 32 .6. 1 The Fourier Cosine Transform . ....

Ngày tải lên: 06/08/2014, 01:21

... is the angle from 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 ... components. • If a · b = 0 then either a and b are orthogonal, (perpendicular), or one of a and b are zero. 26 1 2 1 2 2 4 6 8 10 1 2 Figure 1.11: Plots of f(x) = p(x)/q(x). 1...

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 ... δ − √ x is a decreasing function of x and an increasing function of δ for positive x and δ. Bound this function for fixed δ. Consider any positive δ and . For what values of...

Ngày tải lên: 06/08/2014, 01:21