Page 428 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 419

2
2
(b)Region in the first quadrant bounded by x þ y ¼ 9, the x-axis and the line y ¼ x.
2
2
(c)Interior of ellipse x =25 þ y =16 ¼ 1.
16.41. Express each function in the form uðx; yÞþ ivðx; yÞ, where u and v are real.
2
2
z
(a) z þ 2iz; ðbÞ z=ð3 þ zÞ; ðcÞ e ; ðdÞ lnð1 þ zÞ.
2
2
3
2
Ans. (a) u ¼ x 3xy 2y; v ¼ 3x y y þ 2x
2
x þ 3x þ y 2 3y
(b) u ¼ 2 2 ; v ¼ 2 2
x þ 6x þ y þ 9 x þ 6x þ y þ 9
2
2
(c) u ¼ e x y 2 cos 2xy; v ¼ e x y 2 sin 2xy
y
2
2
1
(d) u ¼ lnfð1 þ xÞ þ y g; v ¼ tan 1 þ 2k ; k ¼ 0; 1; 2; .. .
2 1 þ x
2
2
2
16.42. Prove that (a) lim z ¼ z 0 ; ðbÞ f ðzÞ¼ z is continuous at z ¼ z 0 directly from the definition.
z!x 0
3
2
4
5
16.43. (a)If z ¼ ! is any root of z ¼ 1different from 1, prove that all the roots are 1;!;! ;! ;! .
2
4
3
(b) Show that 1 þ ! þ ! þ ! þ ! ¼ 0.
n
(c)Generalize the results in (a) and (b)to the equation z ¼ 1.
DERIVATIVES, CAUCHY-RIEMANN EQUATIONS
1 dw
16.44. (a)If w ¼ f ðzÞ¼ z þ , find directly from the definition.
z dz
(b) For what finite values of z is f ðzÞ nonanalytic?
2
Ans. ðaÞ 1 1=z ; ðbÞ z ¼ 0
4
16.45. Given the function w ¼ z . (a)Find real functions u and v such that w ¼ u þ iv. (b) Show that the
Cauchy-Riemann equations hold at all points in the finite z plane. (c)Prove that u and v are harmonic
functions. (d)Determine dw=dz.
4
2 2
4
3
Ans: ðaÞ u ¼ x 6x y þ y ; v ¼ 4x y 4xy 2 ðdÞ 4z 3
16.46. Prove that f ðzÞ¼ zjzj is not analytic anywhere.
1
is analytic in any region not including z ¼ 2.
16.47. Prove that f ðzÞ¼
z 2
16.48. If the imaginary part of an analytic function is 2xð1 yÞ,determine (a)the real part, (b)the function.
2
2
2
Ans: ðaÞ y x 2y þ c; ðbÞ 2iz z þ c, where c is real
x
16.49. Construct an analytic function f ðzÞ whose real part is e ðx cos y þ y sin yÞ and for which f ð0Þ¼ 1.
Ans: ze z þ 1
2
16.50. Prove that there is no analytic function whose imaginary part is x 2y.
16.51. Find f ðzÞ such that f ðzÞ¼ 4z 3 and f ð1 þ iÞ¼ 3i.
0
2
Ans: f ðzÞ¼ 2z 3z þ 3 4i
INTEGRALS, CAUCHY’S THEOREM, CAUCHY’S INTEGRAL FORMULAS
ð 3þi
16.52. Evaluate ð2z þ 3Þ dz:
1 2i
2
(a)along the path x ¼ 2t þ 1; y ¼ 4t t 20 @ t @ 1.
(b)along the straight line joining 1 2i and 3 þ i.
(c)along straight lines from 1 2i to 1 þ i and then to 3 þ i.
Ans: 17 þ 19i in all cases