Complex Analysis - Maths KU

44 Chapter 1 Complex Numbers and the Complex Plane
In Problems 35 and 36, use a CAS to solve the given polynomial equation. In
Mathematica the command Solve will find all roots of polynomial equations up to
degree four by means of a formula.
35. z 3 − 4z 2 +10=0 36. z 4 +4iz 2 +10i =0
In Problems 37 and 38, use a CAS to solve the given polynomial equation. The command
NSolve in Mathematica will approximate all roots of polynomial equations
of degree five or higher.
37. z 5 − z − 12=0 38. z 6 − z 4 +3iz 3 − 1=0
Projects
39. Cubic Formula In this project you are asked to investigate the solution of
a cubic polynomial equation by means of a formula using radicals, that is, a
combination of square roots and cube roots of expressions involving the coefficients.
(a) To solve a general cubic equation z 3 + az 2 + bz + c = 0 it is sufficient to
solve a depressed cubic equation x 3 = mx + n since the general cubic
equation can be reduced to this special case by eliminating the term az 2 .
Verify this by means of the substitution z = x − a/3 and identify m and n.
(b) Use the procedure outlined in part (a) to find the depressed cubic equation
for z 3 +3z 2 − 3z − 9=0.
(c) A solution of x 3 = mx + n is given by
�
n
x =
2 +
� � �
2
1/2 1/3 �
n m3
n
− +
4 27
2 −
� � �
2
1/2 1/3
n m3
− .
4 27
Use this formula to solve the depressed cubic equation found in part (b).
(d) Graph the polynomial z 3 +3z 2 − 3z − 9 and the polynomial from the
depressed cubic equation in part (b); then estimate the x-intercepts from
the graphs.
(e) Compare your results from part (d) with the solutions found in part (c).
Resolve any apparent differences. Find the three solutions of z 3 +3z 2 −
3z − 9=0.
(f) Do some additional reading to find geometrically motivated proofs (using a
square and a cube) to derive the quadratic formula and the formula given
in part (c) for the solution of the depressed cubic equation. Why is the
name quadratic formula used when the prefix quad stems from the Latin
word for the number four?
40. Complex Matrices In this project we assume that you have either had some
experience with matrices or are willing to learn something about them.
Certain complex matrices, that is, matrices whose entries are complex numbers,
are important in applied mathematics. An n × n complex matrix A is
said to be:
Hermitian if ĀT = A,
Skew-Hermitian if ĀT = −A,
Unitary if ĀT = A −1 .