Complex Analysis - Maths KU

Chapter 3 Review Quiz 173
11. The Cauchy-Riemann equations can be satisfied at a point z, but the function
f(z) =u(x, y)+iv(x, y) can be nondifferentiable at z.
12. Ifthe function f(z) =u(x, y)+iv(x, y) is analytic at a point z, then necessarily
the function g(z) =v(x, y) − iu(x, y) is analytic at z.
In Problems 13–22, try to fill in the blanks without referring back to the text.
1
13. If f(z) =
z2 +5iz − 4 , then f ′ (z) = .
14. The function f(z) =
1
z2 is not analytic at .
+5iz − 4
15. The function f(z) =(2− x) 3 + i(y − 1) 3 is differentiable at z = .
16. For f(z) =2x 3 +3iy 2 , f ′� x + ix 2� = .
x − 1
17. The function f(z) =
(x − 1) 2 y − 1
2 − i
+(y − 1) (x − 1) 2 2 is analytic in a
+(y − 1)
domain D not containing the point z =1+i. InD, f ′ (z) = .
18. Find an analytic function f(z) = loge not containing the origin.
� x 2 + y 2 + i in a domain D
19. The function f(z) is analytic in a domain D and f(z) =c + iv(x, y), where c
is a real constant. Then f is a in D.
z
20. lim
z→2i
5 − 4iz 4 − 4z 3 + z 2 − 4iz +4
5z4 − 20iz3 − 21z2 = .
− 4iz +4
21. u(x, y) =c1 where u(x, y) =e −x (x sin y − y cos y) and v(x, y) =c2 where
v(x, y) = are orthogonal families.
22. The statement “There exists a function f that is analytic for Re(z) ≥ 1 and is
not analytic anywhere else” is false because .