Answer Problems 1–16 without referring back to the text. Fill in the blank or answer true/false.

1. Re(1 + i)¹⁰ = _____ and Im(1 + i)¹⁰ = _____.
2. If z is a point in the third quadrant, then i is in the _____ quadrant.
3. If z = 3 + 4i, then Re = _____.
4. i¹²⁷ − 5i⁹ + 2i⁻¹ = _____
5. If z = , then |z| = _____.
6. Describe the region defined by 1 ≤ |z + 2| ≤ 3. _____
7. Arg(z + ) = 0 _____
8. If z = , then Arg z = _____.
9. If e^z = 2i, then z = _____.
10. If |e^z| = 1, then z is a pure imaginary number. _____
11. The principal value of (1 + i)⁽²⁺ⁱ⁾ is _____.
12. If f(z) = x² − 3xy − 5y³ + i(4x²y − 4x + 7y), then f(−1 + 2i) = _____.
13. If the Cauchy–Riemann equations are satisfied at a point, then the function is necessarily analytic there. _____
14. f(z) = e^z is periodic with period _____.
15. Ln(−ie³) = _____
16. f(z) = sin(x − iy) is nowhere analytic. _____

In Problems 17–20, write the given number in the form a + ib.

17. i(2 − 3i)²(4 + 2i)
18. 
19. 
20. 4e^πi/3e^−πi/4

In Problems 21–24, sketch the set of points in the complex plane satisfying the given inequality.

21. Im(z²) ≤ 2
22. Im(z + 5i) > 3
23. ≤ 1
24. Im(z) < Re(z)
25. Look up the definitions of conic sections in a calculus text. Now describe the set of points in the complex plane that satisfy the equation |z − 2i| + |z + 2i| = 5.
26. Let z and w be complex numbers such that |z| = 1 and |w| ≠ 1. Prove that

    

In Problems 27 and 28, find all solutions of the given equation.

27. z⁴ = 1 − i
28. 
29. If f(z) = z²⁴ − 3z²⁰ + 4z¹² − 5z⁶, find .
30. Write f(z) = Im(z − 3) + zRe(z²) − 5z in the form f(z) = u(x, y) + iv(x, y).

In Problems 31 and 32, find the image of the line x = 1 in the w-plane under the given mapping.

31. f(z) = x² − y + i(y² − x)
32. f(z) =

In Problems 33–36, find all complex numbers for which the given statement is true.

33. z = z⁻¹
34. 
35. = −z
36. z² = ()²
37. Show that the function f(z) = −(2xy + 5x) + i(x² − 5y − y²) is analytic for all z. Find f′(z).
38. Determine whether the function

    f(z) = x³ + xy² − 4x + i(4y − y³ − x²y)

    is differentiable. Is it analytic?

In Problems 39 and 40, verify the given equality.

39. Ln[(1 + i)(1 − i)] = Ln(1 + i) + Ln(1 − i)
40. = Ln(1 + i) − Ln(1 − i)