4 randall d knight physics for scientists and engineers a strategic approach with modern physics 39

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx
... approximate sin (1) , 13 15 1 + ≈ 0.8 41 6 67 12 0 10 7 To see that this has the required accuracy, sin (1) ≈ 0.8 41 4 71 Solution 3 .19 Expanding the terms in the approximation in Taylor series, ∆x3 ∆x4 ∆x2 f ... Example 4. 4 .1 Consider the partial fraction expansion of + x + x2 (x − 1) 3 The expansion has the form a0 a1 a2 + + (x − 1) (x − 1) x 1 127 The coefficients are (1 + x + x2 )|x =1 = 3, 0! d a1 = (1 + ... − x→0 10 9 x =0 c ln lim x→+∞ 1+ x x = lim x→+∞ = lim x→+∞ = lim x→+∞ = lim ln + 1/ x 1+ x→+∞ 1+ =1 Thus we have lim x→+∞ 1+ 11 0 x x 1 x − x2 1/ x2 x→+∞ = lim x ln 1+ x x ln + x x = e x 1 d It...
  • 40
  • 170
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx
... continuously deformed to C2 on the domain where the integrand is analytic Thus the integrals have the same value 5 14 -4 -2 -2 -4 Figure 11 .2: The contours and the singularities of 3z +1 -6 -4 -2 -2 -4 -6 ... , C2 and C2 C z dz = z3 − C1 z− + √ C2 z− C3 z− + = 2 + 2 + 2 z− √ √ √ z− z− z √ dz e 2 /3 z − e− 2 /3 z √ √ dz 2 /3 z − 9e z − e− 2 /3 z √ √ dz z − e 2 /3 z − e− 2 /3 z− 9 √ z e 2 /3 ... deform C onto C1 and C2 = C + C1 520 C2 -4 C1 C2 -2 C -2 -4 Figure 11.5: The contours for (z +z+ı) sin z z +ız We use the Cauchy Integral Formula to evaluate the integrals along C1 and C2 ...
  • 40
  • 167
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 4 pot

Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 4 pot
... 4x3 To check the theorem, d x x2 d ∆[A(x)] = dx dx x2 x4 2x x x2 = + 2x 4x3 x x = x4 − 2x3 + 4x4 − 2x3 = 5x4 − 4x3 16 .4. 2 The Wronskian of a Set of Functions A set of functions {y1 , y2 , ... Example 16 .4. 1 Consider the the matrix A(x) = x x2 x2 x4 The determinant is x5 − x4 thus the derivative of the determinant is 5x4 − 4x3 To check the theorem, d x x2 d ∆[A(x)] = dx dx x2 x4 2x x ... [¯] = y 9 03 For the same reason, if yp is a particular solution, then yp is a particular solution as well Since the real and imaginary parts of a function y are linear combinations of y and y ,...
  • 40
  • 162
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 1 docx

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 1 docx
... differential equation as ∞ w = c1 z 3 /4 1+ n =1 n 16 ∞ zn n!(n + 1) ! + c2 z 1 /4 n =1 ∞ c1 z 3 /4 + c2 z 1 /4 1+ n =1 23.2 .1 1+ 16 n 16 n zn n!(n + 1) ! zn n!(n + 1) ! Indicial Equation Now let’s ... c2 = (1 − c1 r1 ) r2 We substitute this into the second equation (1 − c1 r1 )r2 = r2 c1 (r1 − r1 r2 ) = − r2 c r1 + 11 81 n c1 = = = − r2 − r1 r2 r1 √ 1 √ √ 1+ 5 √ 1+ √ √ 1+ 5 1 =√ Substitute ... 31 an = 2(n 1) /2 (n − 2)(n − 4) · · · (1) For the even terms, a2 = a4 = 22 a6 = 42 an = 2(n−2)/2 (n − 2)(n − 4) · · · (2) Thus an = 2(n 1) /2 (n−2)(n 4) ··· (1) 2(n−2)/2 (n−2)(n 4) ···(2) 11 83 for...
  • 40
  • 121
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 2 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 2 pps
... + x + x2 /2 + O(x3 ))(x + a2 x2 + a3 x3 + O(x4 )) = (2a2 x + 6a3 x2 ) + (2x + 4a2 x2 ) + (6x + 6(1 + a2 )x2 ) = O(x3 ) = 17 a2 = 4, a3 = 17 y1 = x − 4x2 + x3 + O(x4 ) Now we see if the second ... yields z= t dz = − dt t d d = −t2 dz dt w + d2 d d = −t2 −t2 dz dt dt d d = t4 + 2t3 dt dt 1 24 0 The equation for u is then t4 u + 2t3 u + (2t + 3t2 )(−t2 )u + t2 u = u + −3u + u = t We see that ... − a0 (2n)(2n − 2) · · · · a0 = (−1)n n , n≥0 m=1 2m = (−1)n a2n−1 2n + a2n−3 = (2n + 1)(2n − 1) a2n+1 = − a1 (2n + 1)(2n − 1) · · · · a1 = (−1)n n , n≥0 m=1 (2m + 1) = (−1)n If {w1 , w2 } is...
  • 40
  • 75
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 3 pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 3 pdf
... 10−29 4. 14 × 10 37 2.09 × 10 45 One Term Relative Error 0 .32 03 0.1 044 0.0507 0.0296 0.0192 0.0 135 0.0100 0.0077 0.0061 0.0 049 Three Term Relative Error 0. 649 7 0.0182 0.0020 3. 9 · 10 4 1.1 · 10 4 3. 7 ... = x 3x2 − P2 (x) = 5x − 3x P3 (x) = 35 x4 − 30 x2 + P4 (x) = Expanding cos(πx) in Legendre polynomials cos(πx) ≈ cn Pn (x), n=0 and calculating the generalized Fourier coefficients with the formula ... even for fairly small values of x 24. 3 Integration by Parts Example 24. 3. 1 The complementary error function erfc(x) = √ π 12 63 ∞ e−t dt x 1.75 1.5 1.25 0.75 0.5 0.25 Figure 24. 1: Plot of K0 (x) and...
  • 40
  • 145
  • 0

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

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4 docx
... the eigenvalues as λn and the eigenfunctions as φn for n ∈ Z+ For the moment we assume that λ = is not an eigenvalue and that the eigenfunctions are real-valued We expand the function f (x) ... compute the eigenvalues However, we can often use the formula to obtain information about the eigenvalues before we solve a problem Example 27 .4. 2 Consider the self-adjoint eigenvalue problem −y ... equation formally self-adjoint xy + y + xy = d (xy ) + xy = dx Result 27.2.1 If L = L∗ then the linear operator L is formally self-adjoint Second order formally self-adjoint operators have the form...
  • 40
  • 147
  • 0

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

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 5 pdf
... not a good approximation 13 54 0 .5 -1 0 .4 0.2 -0 .5 0 .5 -1 -0 .5 0 .5 -0 .5 -0.2 -1 -0 .4 Figure 28.7: Three Term Approximation for a Function with Jump Discontinuities and a Continuous Function A ... -1 0 .5 -0 .5 0 .5 1 .5 -1 -0 .5 0 .5 -0 .5 -0 .5 -1 1 .5 -1 Figure 28.3: A Function Defined on the range −1 ≤ x < and the Function to which the Fourier Series Converges bn = = = 3/2 3 f (x) sin −1 5/ 2 ... + for − < x < −1/2 for − 1/2 < x < 1/2 for 1/2 < x < 1 355 0 .5 0.2 0.1 -1 -0 .5 0 .5 0. 25 0.1 -0.1 -0.2 0.1 Figure 28.8: Three Term Approximation for a Function with Continuous First Derivative and...
  • 40
  • 83
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 6 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 6 pps
... Parseval’s theorem for this series to find the value of ∞ 16 n=1 π 1 = n6 π −π ∞ 16 n=1 ∞ n=1 x3 π − x 3 16 = n6 945 6 = n6 945 13 76 ∞ n=1 n6 dx Solution 28.2 We differentiate the partial sum of ... 16 = n4 π n=1 ∞ π x4 dx −π 2π 2π + 16 = n4 n=1 ∞ n=1 4 = n4 90 1375 Now we integrate the series for f (x) = x2 x ξ2 − ∞ π2 3 dξ = n=1 ∞ (−1)n n2 x cos(nξ) dξ x π (−1)n − x =4 sin(nx) 3 n3 n=1 ... = π 4( −1)n = n2 a0 = Thus the Fourier series is ∞ π2 (−1)n x = +4 cos(nx) for x ∈ (−π π) n2 n=1 ∞ n=1 n4 We apply Parseval’s theorem for this series to find the value of ∞ 2π 1 + 16 = n4 π...
  • 40
  • 156
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 7 docx

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 7 docx
... eigenvalues and eigenfunctions for: d4 φ = λφ, dx4 φ(0) = φ (0) = 0, φ(1) = φ (1) = Hint, Solution 144 2 29.5 Hints Hint 29.1 Hint 29.2 Hint 29.3 Hint 29 .4 Write the problem in Sturm-Liouville form to ... = 0, λ = 1 /4 is not an eigenvalue 144 9 Now consider the case λ = 1 /4 A set of solutions is √ (x + 1)(1+ 1 4 )/2 , (x + 1)(1− √ 1 4 )/2 We can write this in terms of the exponential and the logarithm ... p0 y = µy, for a ≤ x ≤ b, α1 y(a) + α2 y (a) = 0, β1 y(b) + β2 y (b) = 0, where the pj are real and continuous and p2 > on [a, b], and the αj and βj are real can be written in the form of a regular...
  • 40
  • 152
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 8 potx

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 8 potx
... s 146 8 From part (a) we know that there are only positive eigenvalues The general solution of the differential equation is φ = c1 cos(λ1 /4 x) + c2 cosh(λ1 /4 x) + c3 sin(λ1 /4 x) + c4 sinh(λ1 /4 x) ... conditions c1 sin(λ1 /4 ) + c2 sinh(λ1 /4 ) = −c1 λ1/2 sin(λ1 /4 ) + c2 λ1/2 sinh(λ1 /4 ) = We see that sin(λ1 /4 ) = The eigenvalues and eigenfunctions are λn = (nπ )4 , φn = sin(nπx), 146 9 n ∈ N Chapter ... r≤ √ Thus the smallest zero of J0 (x) is less than or equal to ≈ 2 .44 94 (The smallest zero of J0 (x) is approximately 2 .40 483 .) (1 Solution 29.9 We assume that < l < π Recall that the solution...
  • 40
  • 110
  • 0

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 9 docx

Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 9 docx
... the answer for ν = 0? Hint, Solution 1 49 9 Exercise 31.12 Find the inverse Laplace transform of ˆ f (s) = s3 − 2s2 +s−2 with the following methods ˆ Expand f (s) using partial fractions and then ... transform to find y(t) We could expand the right side in partial fractions and then use a table of Laplace transforms Since the function is analytic except for 15 29 isolated singularities and vanishes ... Laplace transform of y (s) by first finding its partial fraction expansion ˆ s/3 s/3 s − + +1 s +4 s +1 s/3 4s/3 + =− s +4 s +1 y(t) = − cos(2t) + cos(t) 3 y (s) = ˆ s2 Example 31 .4. 3 Consider...
  • 40
  • 109
  • 0

Xem thêm

Từ khóa: hướng dẫn autocad 11111111111111111111111111111122 Đề thi toán tuyển sinh vào lớp 10 các tỉnhthành phố ( có đáp án ) năm học 20172018Nghiên cứu tác động kích thích internet marketing tới hành vi người tiêu dùng – trường hợp công ty cổ phần phát hành sách thành phố hồ chí minh (FAHASA)Một số bài toán cực trị cho đa thức nhiều biến (LV thạc sĩ)CHIẾN LƯỢC PHÁT TRIỂN TRƯỜNG THCS GHỀNH RÁNG GIAI ĐOẠN 2010 2020giáo án thực hành lái xeBÀI THU HOẠCH THỰC TẾ Lớp trung cấp lý luận chính trị hành chínhĐề Tài: MỘT SỐ BIỆN PHÁP GIÚP TRẺ 2436 THÁNG HỨNG THÚ LÀM QUEN VĂN HỌC – THỂ LOẠI TRUYỆNLời giải đề thi Toán THPT Quốc gia năm 2017 mã đề 101Khoá luận tốt nghiệp đại học chuyên ngành khoa học môi trường đánh giá hiện trạng ô nhiễm nước thải hầm lò mỏ than tại công ty TNHH MTV 790Report giản đồ pha của nước (h2o) và carbon (c)Bộ đề minh họa và đề thi thử theo cụm THPTQG Hóa họcPhát triển nguồn nhân lực đáp ứng yêu cầu công nghiệp hóa, hiện đại hóa ở thành phố Việt Trì, tỉnh Phú Thọ (LV thạc sĩ)Hoàn thiện quản lý ngân sách nhà nước trên địa bàn phường Thọ Sơn, thành phố Việt Trì, tỉnh Phú Thọ (LV thạc sĩ)Nâng cao chất lượng nguồn nhân lực tại Công ty Cổ phần thương mại Thái Hưng (LV thạc sĩ)Nghiên cứu một số thuật toán lọc thư rác và ứng dụng trong lọc email nội bộ của viễn thông tỉnh Bắc Kạn (LV thạc sĩ)Chất lượng công chức ngành lao động phúc lợi xã hội thủ đô viêng Chăn, nước cộng hòa dân chủ nhân dân Lào (LV thạc sĩ)Hoạt động của văn phòng tỉnh Bolykhamxay nước cộng hòa dân chủ nhân dân Lào (LV thạc sĩ)Năng lực chủ chốt cán bộ chính quyền cấp xã ở huyện Hà Trung, tỉnh Thanh Hóa hiện nay (LV thạc sĩ)Quản lý nhà nước về nông nghiệp tỉnh Luông Pha Bang, nước Cộng hòa dân chủ nhân dân Lào (LV thạc sĩ)
Đăng ký
Đăng nhập