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Find the modulus, argument, principal value of argument, and polar form of the given complex number: (i) ¹-3 (ii) (3) (iii) -100 (iv) -3i. (2+1)²

Find the modulus, argument, principal value of argument, and polar form of the given complex number: (i) ¹-3 (ii) (3) (iii) -100 (iv) -3i. (2+1)²

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 22E
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1. Find the modulus, argument, principal value of argument, and polar form of the given complex number: (6) (i) 유 (ii) -100 (iv) -3i. 2. Find all the roots for all the values of the following: (i) Fourth roots of –2/3 – 2i (ii) The values of iš. 3. Show that the following: (i) Discuss the analyticity of the function f(z) = ry°(x+ iy) + x² + y*, iƒz # 0, f(0) = 0 at orgin. (ii) |z|² and z are not analytic at any point in the complex plane. 4. Find the analytic function f(2) = u+ iv given the following: (First verify that they are harmonic functions) (i) v = e (xsin(y) + ycos(y)) (ii) u = -r³cos(30) (iii) u – v = (x – y)(x² + 4ry + y²). 5. (a) Find the critical point of the following transformations: (i) w = 24 – 1 (ii) w² = z(z – 3) (b) Find the fixed point of the following transformations: (i) w = (ii) w = 6. (i) Find the image of the triangle region in the z-plane bounded by the lines x = 0, y = 0 and r+y = 1 under the transformation uw = (1+2i)z + (1+ i). (ii) Find the image of the region bounded by 1 <r < 2 and 1<y<2 under the transformation w = 22. 7. (i) Find the transformation which maps the triangular region 0 < arg(2) < into the unit circle |w| < 1. (ii) Find the image of the grid which consists of the vertical line segments r = 1 , 2 and 3, * <ys, and the horizontal line segments y = , 7,0, 5 and , 1 < 1 < 3 under w = e*. 8. (i) Find the cross ratio of z1 = 0, 22 = 1, 23 = i and z4 = ∞. (ii) Find the bilinear transformation which maps the points z = -2, 0, 2 respectively onto the points w = 0, i, - i. 9. (i) Find the bilinear transformation which maps the points z = i, – 1, 1 respectively onto the points w = 0, 1, . (ii) Prove that the transformation w = maps the upper half of the z-plane into the upper half of the w-plane. What is the image of the circle |2| =1 under this transformation. 10. (i) Expand the function f(2) = sin(2) in a Taylor's series at z = 7. (ii) Find the Laurent's series of f(z) = for (a) |2| < 2, (b) |2| > 3 and (c) 2 < |2| < 3. 2-1 9+zs+ %3D
Transcribed Image Text:1. Find the modulus, argument, principal value of argument, and polar form of the given complex number: (6) (i) 유 (ii) -100 (iv) -3i. 2. Find all the roots for all the values of the following: (i) Fourth roots of –2/3 – 2i (ii) The values of iš. 3. Show that the following: (i) Discuss the analyticity of the function f(z) = ry°(x+ iy) + x² + y*, iƒz # 0, f(0) = 0 at orgin. (ii) |z|² and z are not analytic at any point in the complex plane. 4. Find the analytic function f(2) = u+ iv given the following: (First verify that they are harmonic functions) (i) v = e (xsin(y) + ycos(y)) (ii) u = -r³cos(30) (iii) u – v = (x – y)(x² + 4ry + y²). 5. (a) Find the critical point of the following transformations: (i) w = 24 – 1 (ii) w² = z(z – 3) (b) Find the fixed point of the following transformations: (i) w = (ii) w = 6. (i) Find the image of the triangle region in the z-plane bounded by the lines x = 0, y = 0 and r+y = 1 under the transformation uw = (1+2i)z + (1+ i). (ii) Find the image of the region bounded by 1 <r < 2 and 1<y<2 under the transformation w = 22. 7. (i) Find the transformation which maps the triangular region 0 < arg(2) < into the unit circle |w| < 1. (ii) Find the image of the grid which consists of the vertical line segments r = 1 , 2 and 3, * <ys, and the horizontal line segments y = , 7,0, 5 and , 1 < 1 < 3 under w = e*. 8. (i) Find the cross ratio of z1 = 0, 22 = 1, 23 = i and z4 = ∞. (ii) Find the bilinear transformation which maps the points z = -2, 0, 2 respectively onto the points w = 0, i, - i. 9. (i) Find the bilinear transformation which maps the points z = i, – 1, 1 respectively onto the points w = 0, 1, . (ii) Prove that the transformation w = maps the upper half of the z-plane into the upper half of the w-plane. What is the image of the circle |2| =1 under this transformation. 10. (i) Expand the function f(2) = sin(2) in a Taylor's series at z = 7. (ii) Find the Laurent's series of f(z) = for (a) |2| < 2, (b) |2| > 3 and (c) 2 < |2| < 3. 2-1 9+zs+ %3D
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