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

... it. I planned it that way. 17 05 36 .5 Solutions Solution 36.1 For y = −1, the equation is parabolic. For this case it is already in the canonical form, u xx = 0. For y = −1, the equation is elliptic. ... 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 =...

Ngày tải lên: 06/08/2014, 01:21

... 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 ... [−1, 1] and −1 ≤ cos(x 0 ) ≤ 1, the approximation sin x ≈ x − x 3 6 + x 5 120 has a maximum error of 1 50 40 ≈ 0.000198. Using this polynomial to approximate sin(1), 1 − 1 3...

Ngày tải lên: 06/08/2014, 01:21

... − 5 14 x(2 − x). This one term approximation and the exact solution are plotted in Figure 48 .7. The value of the functional is J = − 10 21 ≈ −0 .47 619. 0 .5 1 1 .5 2 -0. 35 -0.3 -0. 25 -0.2 -0. 15 -0.1 -0. 05 Figure ... 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 Figure 48 .5: One Term Approximation and Exact Solution. n = 1. We...

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... 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 ... from (1, 1) to (4, 2) and the vector from (1, 1) to (2, 3). The second triangle is defined by the vector from (3, 7) to (4, 2) and the vector from (3, 7) to (2, 3). (See Figure 2. 15. ) T...

Ngày tải lên: 06/08/2014, 01:21

... 3.11. -1 -0 .5 0 .5 1 0 .5 1 1 .5 2 2 .5 -1 -0 .5 0 .5 1 0 .5 1 1 .5 2 2 .5 -1 -0 .5 0 .5 1 0 .5 1 1 .5 2 2 .5 -1 -0 .5 0 .5 1 0 .5 1 1 .5 2 2 .5 Figure 3.11: Four Finite Taylor Series Approximations of e x Note that for ... through the points (x, y(x)) and (x + ∆x, y(x + ∆x)). See Figure 3 .5. y x ∆ y ∆ x Figure 3 .5: The increments ∆x and ∆y. 56 -10 -5 5 10 -2...

Ngày tải lên: 06/08/2014, 01:21

... 4 −63 − ı16 = 3 + 4 −63 − ı16 −63 + ı16 −63 + ı16 = − 253 42 25 − ı 2 04 42 25 2 15 2. (1 − ı) 7 =  (1 − ı) 2  2 (1 − ı) 2 (1 − ı) = (−ı2) 2 (−ı2)(1 − ı) = ( 4) (−2 −ı2) = 8 + ı8 Solution 6 .5 1.  z z  =  x ... 2 −10 e −ı10π/3 = 1 10 24  cos  − 10π 3  + ı sin  − 10π 3  = 1 10 24  cos  4 3  − ı sin  4 3  = 1 10 24  − 1 2 + ı √ 3 2  = − 1 2 048 + ı √ 3 2 048...

Ngày tải lên: 06/08/2014, 01:21

... Z z = 4m 1 + 4n , m, n ∈ Z There are an infinite set of rational numbers for which ı z has 1 as one of its values. For example, ı 4 /5 = 1 1 /5 =  1, e ı2π /5 , e 4 /5 , e ı6π /5 , e ı8π /5  7.8 ... cuts and define a branch so that z = 4 does not lie on a cut, and such that w = ı6 when z = 4. Hint, Solution Exercise 7. 24 Give the number of branches and locations of th...

Ngày tải lên: 06/08/2014, 01:21

... coordinates.    e z 2    =    e (x+ıy) 2    =    e x 2 −y 2 +ı2xy    = e x 2 −y 2 3 04 25 50 75 100 -1 1 Figure 7 .43 : The values of ı ı . See Figure 7 .44 for a plot. -40 -20 20 -10 -5 5 10 Figure 7 .44 : The values of log ((1 + ı) ıπ ). 317 Solution 7.13 cot z = 47 ( e ız + e −ız ) ... ı  1 /4 =  2 e −ıπ/6  1 /4 = 4 √ 2 e −ıπ/ 24 1 1 /4 = 4 √ 2 e ı(π...

Ngày tải lên: 06/08/2014, 01:21

... = 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2 u ∂θ 2 = 0. Therefore u is harmonic and is the real part of some analytic function. Example 9.3.9 Find an analytic function f(z) whose real part is u(r, θ) = r (log r cos θ −θ sin θ) . 44 7 The complex ... real and imaginary parts. u r = 1 r v θ , v r = − 1 r u θ u r = 1 r v θ , u θ = −rv r Solution 8. 15 Since w is analytic, u and v satisfy the...

Ngày tải lên: 06/08/2014, 01:21

... + e ıθ  n ı e ıθ dθ =       ( ı2+ e ıθ ) n+1 n+1  2π 0 for n = −1  log  ı2 + e ıθ  2π 0 for n = −1 = 0 3. We parameterize the contour and do the integration. z = r e ıθ , r = 2 −sin 2  θ 4  , θ ∈ [0 . . . 4 ) 48 7 Solution 10.6  C f(z) ... a formula for writing the analytic function in terms of its real part, u(r, θ). With the same method, we can find how to write...

Ngày tải lên: 06/08/2014, 01:21