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11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 6z +z (b) Evaluate dz where C is the circle z-1= 1. z -1 C.

11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 6z +z (b) Evaluate dz where C is the circle z-1= 1. z -1 C.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please can u send the solution of these problems please it will be great help plzz I request u to send all the question solutions plzz .you can send the solution one by one plzzzz 

11. a) Find the analytic function f(z)=u+iv, if u-v = (x – y\x² + 4.xy+ y² ). 15. A homogeneous rod of conducting material of length 100 cm has its ends kept at 0SxS 50 (11 b) Evaluate sin z + coS TZ -dz where C is the circle z= 3. x, (z – 1)(z – 2) zero temperature and the temperature initially is u (x, 0)= 100-x 50sIS100 Find the temperature u(x,1) at any time. 12. a) Find Laurent's expansion of 7z- 2 z(z+1)(z – 2) x² b) Evaluate +1)x* + 4) f(2) = in the region 1<|z+ 1|< 3. 16. a) Find the bilinear transformation which maps the points z = 1, i,-1 onto the points 'E > + z|> 1 w = i,0,-i. 8. dx. b) Evaluate dz where C is the circle |z – 2| = ÷. (z- 1Xz- 2) 13. If f(x)=cos x), expand f(x)as a Fourier series in the interval (-,7). -2- 2e2* +3x y. %3D 17. a) Solve 14. a) Solve (x? - y² - z² )p+ 2xyq = 2xz. b) Solve 2z + p² + qy+ 2y = 0 by using Charpit's method. %3D b) Find the complete solution of z (p² +q° )= x² + y², 3 %3D 13 a) Find the Fourier series expression of f(x) = x + x, -T S X ST. 1 Deduce that 1+ 1 1 22 32 42 b) Obtain the half range Fourier sine series for f(x) = 2x – 1,0 < x < 1. 14 a) Obtain the general solution of 2xzp + 2yzq + x² + y - z? = 0. b) Solve p + q = pq by Charpit's method. 11 a) Show that the function u(x, y) = 4xy - 3x+2 is harmonic and find its conjugate %3D harmonic function v(x, y). 15 Solve the Laplace equation- ,2 dy = 0 for a rectangular plate subject to the following z-1 b) Apply Cauchy's integral formula to evaluate dz, where C is |z-i| = 2. (z+1)2 (z- 2) conditions. U(x, 0) = u(x, 1) = 0, u(0, y) = y and ди - (1, y) = -5 z2 -1 12 a) Expand f(z) = in the regions (i) Iz| < 2 and (ii) 2 < |z| < 3. (z+2)(z+3) 16 a) Find the analytic function f(z) such that Re[f (z)] = 3x² – 4y – 3y² and f(1+i) = 0. %3D %3D 2x b) Evaluate| 1+2 cos0 de. b) Find the bilinear transformation which maps the points z = 1, i, -1 into w = i, 0, -i and hence find the image of |z| <.1. 5+4cos0 17 a) Solve [2D² – 5DD' + 3(D')² +D- D'] z = e*y. TC Cos cos cos 8 b) Show that |x| = 1-- -2 < x < 2. %3D 32 52 TC 1604
Transcribed Image Text:11. a) Find the analytic function f(z)=u+iv, if u-v = (x – y\x² + 4.xy+ y² ). 15. A homogeneous rod of conducting material of length 100 cm has its ends kept at 0SxS 50 (11 b) Evaluate sin z + coS TZ -dz where C is the circle z= 3. x, (z – 1)(z – 2) zero temperature and the temperature initially is u (x, 0)= 100-x 50sIS100 Find the temperature u(x,1) at any time. 12. a) Find Laurent's expansion of 7z- 2 z(z+1)(z – 2) x² b) Evaluate +1)x* + 4) f(2) = in the region 1<|z+ 1|< 3. 16. a) Find the bilinear transformation which maps the points z = 1, i,-1 onto the points 'E > + z|> 1 w = i,0,-i. 8. dx. b) Evaluate dz where C is the circle |z – 2| = ÷. (z- 1Xz- 2) 13. If f(x)=cos x), expand f(x)as a Fourier series in the interval (-,7). -2- 2e2* +3x y. %3D 17. a) Solve 14. a) Solve (x? - y² - z² )p+ 2xyq = 2xz. b) Solve 2z + p² + qy+ 2y = 0 by using Charpit's method. %3D b) Find the complete solution of z (p² +q° )= x² + y², 3 %3D 13 a) Find the Fourier series expression of f(x) = x + x, -T S X ST. 1 Deduce that 1+ 1 1 22 32 42 b) Obtain the half range Fourier sine series for f(x) = 2x – 1,0 < x < 1. 14 a) Obtain the general solution of 2xzp + 2yzq + x² + y - z? = 0. b) Solve p + q = pq by Charpit's method. 11 a) Show that the function u(x, y) = 4xy - 3x+2 is harmonic and find its conjugate %3D harmonic function v(x, y). 15 Solve the Laplace equation- ,2 dy = 0 for a rectangular plate subject to the following z-1 b) Apply Cauchy's integral formula to evaluate dz, where C is |z-i| = 2. (z+1)2 (z- 2) conditions. U(x, 0) = u(x, 1) = 0, u(0, y) = y and ди - (1, y) = -5 z2 -1 12 a) Expand f(z) = in the regions (i) Iz| < 2 and (ii) 2 < |z| < 3. (z+2)(z+3) 16 a) Find the analytic function f(z) such that Re[f (z)] = 3x² – 4y – 3y² and f(1+i) = 0. %3D %3D 2x b) Evaluate| 1+2 cos0 de. b) Find the bilinear transformation which maps the points z = 1, i, -1 into w = i, 0, -i and hence find the image of |z| <.1. 5+4cos0 17 a) Solve [2D² – 5DD' + 3(D')² +D- D'] z = e*y. TC Cos cos cos 8 b) Show that |x| = 1-- -2 < x < 2. %3D 32 52 TC 1604
11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 2 6z2 +z (b) Evaluate 02 dz where C is the circle z-1= 1. z2 -1 | 12 (a) Expand f(z)= 1 in a Laurent series valid for (i)1</z| <3 (ii) z>3 (z +1)(z+ 3) dx (b) Evaluate „(l+x²)? 14 (a) Solve z(p-q)=z²+(x+y)² (b) Solve 2(z + xp + yp) =yp2 by using Charpits method. du 15 Find the solution of the heat equation a²u 0<x<l,t>0 subject to the condition at u(0, t) = 0 = u(?, t) and u(x, 0) = &x - x². Əx² 16 (a) Find the bilinear transformation which maps the points 1, i, -1 onto the points 604 0,1,00. (b) Evaluate (cos(2°) where C is the circle z=2 cos(z') (z +1)? 17 (a) Solve (D² + DD' – 6D² ) = y sin x - (b) Find the complete and singular integrals of z = px + qy - 2 pq
Transcribed Image Text:11 (a) Find the analytic function f (z) = u + iv where u(x, y) =e* (xcosy - ysiny). 2 6z2 +z (b) Evaluate 02 dz where C is the circle z-1= 1. z2 -1 | 12 (a) Expand f(z)= 1 in a Laurent series valid for (i)1</z| <3 (ii) z>3 (z +1)(z+ 3) dx (b) Evaluate „(l+x²)? 14 (a) Solve z(p-q)=z²+(x+y)² (b) Solve 2(z + xp + yp) =yp2 by using Charpits method. du 15 Find the solution of the heat equation a²u 0<x<l,t>0 subject to the condition at u(0, t) = 0 = u(?, t) and u(x, 0) = &x - x². Əx² 16 (a) Find the bilinear transformation which maps the points 1, i, -1 onto the points 604 0,1,00. (b) Evaluate (cos(2°) where C is the circle z=2 cos(z') (z +1)? 17 (a) Solve (D² + DD' – 6D² ) = y sin x - (b) Find the complete and singular integrals of z = px + qy - 2 pq
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