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FUNCTIONS, LIMITS, CONTINUITY 16.39. Describe the locus represented by (a) jz 2 3ij 5; b jz 2j 2jz 1j; Construct a gure in each case. Ans. a Circle x 2 2 y 3 2 25, center 2; 3 , radius 5. (b) Circle x 2 2 y2 4, center 2; 0 , radius 2. (c) Branch of hyperbola x2 =9 y2 =16 1, where x A 3. 16.40. Determine the region in the z plane represented by each of the following:  (a) jz 2 ij A 4; b jzj @ 3; 0 @ arg z @ ; c jz 3j jz 3j < 10. 4 Construct a gure in each case. Ans. (a) Boundary and exterior of circle x 2 2 y 1 2 16. c jz 5j jz 5j 6.
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(b) Region in the rst quadrant bounded by x2 y2 9, the x-axis and the line y x. (c) Interior of ellipse x2 =25 y2 =16 1. 16.41. Express each function in the form u x; y iv x; y , where u and v are real. (a) z2 2iz; Ans. b z= 3 z ;
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(a) u x 3xy 2y; v 3x y y2 2x (b) u x2 3x y2 3y ;v 2 x2 6x y2 9 x 6x y2 9
(c) u ex
cos 2xy; v ex
sin 2xy y 2k; k 0; 1; 2; . . . 1 x
(d) u 1 lnf 1 x 2 y2 g; v tan 1 2 16.42. Prove that (a) lim z2 z2 ; 0
z!x0
b f z z2 is continuous at z z0 directly from the de nition.
16.43. (a) If z ! is any root of z5 1 di erent from 1, prove that all the roots are 1; !; !2 ; !3 ; !4 . (b) Show that 1 ! !2 !3 !4 0. (c) Generalize the results in (a) and (b) to the equation zn 1. DERIVATIVES, CAUCHY-RIEMANN EQUATIONS 1 dw directly from the de nition. 16.44. (a) If w f z z , nd z dz (b) For what nite values of z is f z nonanalytic Ans. a 1 1=z2 ; b z 0 16.45. Given the function w z4 . (a) Find real functions u and v such that w u iv. (b) Show that the Cauchy-Riemann equations hold at all points in the nite z plane. (c) Prove that u and v are harmonic functions. (d) Determine dw=dz. Ans: a u x4 6x2 y2 y4 ; v 4x3 y 4xy2 d 4z3 16.46. Prove that f z zjzj is not analytic anywhere. 16.47. Prove that f z 1 is analytic in any region not including z 2. z 2 (a) the real part, (b) the function.
16.48. If the imaginary part of an analytic function is 2x 1 y , determine Ans: a y2 x2 2y c; b 2iz z2 c, where c is real
16.49. Construct an analytic function f z whose real part is e x x cos y y sin y and for which f 0 1. Ans: ze z 1 16.50. Prove that there is no analytic function whose imaginary part is x2 2y. 16.51. Find f z such that f 0 z 4z 3 and f 1 i 3i. Ans: f z 2z2 3z 3 4i INTEGRALS, CAUCHY S THEOREM, CAUCHY S INTEGRAL FORMULAS 3 i 2z 3 dz: 16.52. Evaluate
1 2i
(a) along the path x 2t 1; y 4t2 t 2 0 @ t @ 1. (b) along the straight line joining 1 2i and 3 i. (c) along straight lines from 1 2i to 1 i and then to 3 i. Ans: 17 19i in all cases
16.53. Evaluate
FUNCTIONS OF A COMPLEX VARIABLE
[CHAP. 16
z2 z 2 dz, where C is the upper half of the circle jzj 1 tranversed in the positive sense.
Ans:
14=3 z; , where C is the circle 2z 5 a 0; b 5i=2
16.54. Evaluate Ans:
(a) jzj 2;
b jz 3j 2:
16.55. Evaluate
z2 dz, where C is: (a) a square with vertices at 1 i; 1 i; 3 i; 3 i; C z 2 z 1 p (b) the circle jz ij 3; (c) the circle jzj 2. Ans: a 8i=3 b 2i cos z dz; b C z 1 a 2i b ie=3 (a) c 2i=3 ez z dz where C is any simple closed curve enclosing z 1. 4 C z 1
16.56. Evaluate Ans:
16.57. Prove Cauchy s integral formulas. [Hint: Use the de nition of derivative and then apply mathematical induction.] SERIES AND SINGULARITIES 16.58. For what values of z does each series converge a Ans:
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