Question

Complex Analysis:

Use Rouche's Theorem and Argument Principle to determine the number of roots of p(z)=z^4-2z^3+13z^2-2z+36 which lie in each quadrant of the plane.

(Hint: Consider |z|=5 and |z|=1)

Answer #1

Complex Variable
Evaluate the following:
A) ∫_∣z−i∣=2 (2z+6)/(z^2+4) dz
B) ∫_∣z∣=2
1/((z−1)^2(z−3)) dz ( details
please)
C) ∫_∣z∣=1 e^(4/z) dz

1.
a)
Express z = −i−sqrt(3) in the form r cis θ, where θ= Argz, and
then use de Moivre’s theorem to find the two square roots of
−4i.
b) Consider:
i) p(z)=iz^2+z^3+2iz−2z^2+2z. Given that z=2−2i is a zero of
this polynomial, find all of its zeros.
ii) p(z)=z^3−2z^2+9z−18. Factorise into linear factors.

Use Inclusion-Exclusion Principle to find the number of
permutations of
the multiset {1, 2, 3, 4, 4, 5, 5, 6, 6} such that any two
identical integers are not adjacent.

Solve each system by elimination.
1) -x-5y-5z=2
4x-5y+4z=19
x+5y-z=-20
2) -4x-5y-z=18
-2x-5y-2z=12
-2x+5y+2z=4
3) -x-5y+z=17
-5x-5y+5z=5
2x+5y-3z=-10
4) 4x+4y+z=24
2x-4y+z=0
5x-4y-5z=12
5) 4r-4s+4t=-4
4r+s-2t=5
-3r-3s-4t=-16
6) x-6y+4z=-12
x+y-4z=12
2x+2y+5z=-15

Consider the line which passes through the point P(4, 5, 4), and
which is parallel to the line x=1+3t, y=2+6t, z=3+1t
Find the point of intersection of this new line with each of the
coordinate planes:
xy-plane: ( , , )
xz-plane: ( , , )
yz-plane: ( , , )

1. A plane passes through A(1, 2, 3), B(1, -1, 0) and
C(2, -3, -4). Determine vector and parametric equations of
the plane. You must show and explain all steps for full marks. Use
AB and AC as your direction vectors and point A as your starting
(x,y,z) value.
2. Determine if the point (4,-2,0) lies in the plane with vector
equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1,
2).

The complex function f(z) = 1/(z^4 - 1) has poles at +-1 and
+-i, which may or may not contribute to the closed curve integral
around C of f(z)dz. In turn, the closed curve C that you use
depends on the 2nd letter of your first name! Specifically, convert
that letter to its numerical position in the Roman alphabet (A=1,
B=2, ..., Z=26), then divide by 4. Don't worry about fractions,
just save the REMAINDER which will be an integer...

Q
P
Q
TC
1
$36
1
$20
2
$32
2
$32
3
$28
3
$50
4
$24
4
$80
5
$20
5
$120
Please explain each answer
30. Consider Exhibit 3. What is the profit maximizing output for
a monopolist that does not price discriminate?
A) 1 unit
B) 2 units
C) 3 units
D) 4 units
E) 5 units
31. What is the profit maximizing price for a monopolist that
does not price discriminate?
a. $36
b. $32...

Use
Gaussian Elimination to solve and show all steps:
1. (x+4y=6)
(1/2x+1/3y=1/2)
2. (x-2y+3z=7)
(-3x+y+2z=-5)
(2x+2y+z=3)

1. Determine whether the lines are parallel, perpendicular or
neither. (x-1)/2 = (y+2)/5 = (z-3)/4 and (x-2)/4 = (y-1)/3 =
(z-2)/6
2. A) Find the line intersection of vector planes given by the
equations -2x+3y-z+4=0 and 3x-2y+z=-2
B) Given U = <2, -3, 4> and V= <-1, 3, -2> Find a. U
. V b. U x V

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