Complex Analysis - Maths KU

42 Chapter 1 Complex Numbers and the Complex Plane
EXERCISES 1.6 Answers to selected odd-numbered problems begin on page ANS-6.
In Problems 1–6, solve the given quadratic equation using the quadratic formula.
Then use (5) to factor the polynomial.
1. z 2 + iz − 2=0 2. iz 2 − z + i =0
3. z 2 − (1 + i)z +6− 17i =0 4. z 2 − (1 + 9i)z − 20 + 5i =0
5. z 2 +2z − √ 3i =0
6. 3z 2 +(2− 3i)z − 1 − 3i =0[Hint: See Problem 15 in Exercises 1.4.]
In Problems 7–12, express the given complex number in the exponential form
z = re iθ .
7. −10 8. −2πi
9. −4 − 4i 10.
2
1+i
11. (3 − i) 2
12. (1 + i) 20
In Problems 13–16, find solutions of the given homogeneous differential equation.
13. y ′′ − 4y ′ +13y =0 14. 3y ′′ +2y ′ + y =0
15. y ′′ + y ′ + y =0 16. y ′′ +2y ′ +4y =0
In Problems 17 and 18, find the steady-state charge qp(t) and steady-state current
ip(t) for the LRC -series circuit described by the given differential equation. Find the
complex impedance and impedance of the circuit. Use the real method or complex
method discussed on page 41 as instructed.
17. 0.5 d2q +3dq
dt2 dt +12.5q = 10 cos 5t 18. d2q +2dq +2q = 100 sin t
dt2 dt
Focus on Concepts
19. Discuss how (3) can be used to find the four roots of z 4 − 2z 2 +1− 2i =0.
Carry out your ideas.
20. If z1 is a root of a polynomial equation with real coefficients, then its conjugate
z2 =¯z1 is also a root. Prove this result in the case of a quadratic equation
az 2 + bz + c = 0, where a =0,b, and c are real. Start with the properties of
conjugates given in (1) and (2) of Section 1.1.
In Problems 21 and 22, use Problem 20 and (5) of this section to factor the given
quadratic polynomial if the indicated complex number is one root.
21. 4z 2 +12z +34=0; z1 = − 3 5
+
2 2 i
22. 5z 2 − 2z +4=0; z1 = 1
5 +
√
19
5 i
23. (a) Find a quadratic polynomial equation for which 2 − i is one root.
(b) Is your answer to part (a) unique? Elaborate in detail.
24. If z1 is a root of a polynomial equation with nonreal coefficients, then its conjugate
z2 =¯z1 is not a root. Prove this result in the case of a quadratic equation
az 2 + bz + c = 0, where at least one of the coefficients a �= 0,b, orc is not real.
Use your work from Problem 20 and indicate at what point out we can make
this conclusion.