MA437/MA537 Spring 2014
Review Exercises TEST 2
1. Cauchy-Riemann equations. State the corresponding theorem. Where is
f (z) = z 2 + 2z differentiable? Where is it analytic?
2. At what point (if any) is the function f (z) = |z|2 + 3z differentiable? Where
is it analytic?
3. Prove that the function f (z) = Re z is not differentiable for all z.
4. At what points (if any) is the function
f (z) = |z|2 − 3z
differentiable? Where is it analytic?
5. Show that f (z) = Rez − z is nowhere differentiable.
6. Show that 3z + z is nowhere differentiable.
7. At what points (if any) is the function
f (z) = x2 − 2yi
differentiable? Where is it analytic?
8. Give the following definition: f (z) is differentiable at a point z0
9. At what points (if any) is the function
f (z) =
3z − 1
z + 5i
differentiable? Where is it analytic?
10. Find the derivatives of the following functions
11. f (z) =
(z + 2i)
(z − 2i)
12. f (z) = (z 2 − 13)5 (4z 2 + 3iz)3
For each of the following determine the points at which a) the function is differentiable b) the function is analytic
13. f (z) =
z2 − 1
z−i
14. f (z) =
iz 2 + z
z+1
15. f (z) = z − z
16. f (z) = |z|2
17. f (z) = x2 − y 2
18. f (z) = 2xy + xi
19. Verify that the following function is entire
f (z) = z 3 − 2z
20. Verify that the following function is entire
f (z) = ez
2 −3z
21. Show that
e2iθ = e−2iθ
22. Cauchy-Riemann equations in polar coordinates. State and prove the corresponding result. Give examples.
23. Prove that
|ez | ≤ e|z|
24. Give a brief sketch of the strategy we used to prove the formula
ez 6= 0.
25. Give a brief sketch of the strategy we used to prove the formula
ez+2πki = ez ,
where k is an integer.
26. State definitions and prove properties of the trigonometric functions sin z and
cos z
27. Suppose that a function f = u + vi is analytic in some domain G. Prove that
u and v are harmonic in G.
28. Prove that a function Logz is analytic in domain C \ (−∞, 0].
29. True or false
Log (−1 + i)2 = 2Log (−1 + i)
30. Find all values of z such that
Log z =
π
i
2
31. State definitions and prove properties of the exponential function ez .
32. Prove that if z 2 = z 2 , then z is either real or pure imaginary.
33. Let z be a complex number such that Imz > 0. Prove that
1
Im < 0.
z
34. Verify that the function u is harmonic, and then find a harmonic conjugate
of u
1) u = ex sin y
2) u = xy − x + y
3) u = x3 − 3xy 2 + y
35. Write each of the following numbers in the form a + bi
1) exp(2 + πi/4)
2) sin(2i)
3) cos(1 − i)
4) exp(1 + i3π)/ exp(−1 + iπ/2)
36. Find all values of z such that
√
(a) ez = −1; (b) ez = 1 − 3i; (c) exp(2z + 1) = 1
37. Evaluate each of the following
1) log i
2) log(1 − i)
3) log(−i)
√
4) log( 3 + i)
5) log(2 − 2i)
38. Find all roots of the equation
1) log z = (π/3)i + 2πki
2) log(z − 1) = iπ/2 + 2πki
39. Prove that f = u + iv is analytic in a domain D if and only if v is a harmonic
conjugate of u.
40. State and prove properties of log z.
41. State definitions and prove properties of the hyperbolic functions sinh z and
cosh z