Question

Use Rouch´e’s Theorem to determine the number of solutions of equation z^3 = 3iz−1 in the annulus {z : 1 < |z| < 2}.

Answer #1

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)

Use a graphing utility to determine the number of real solutions
of the quadratic equation. 1/3x^2 − 5x + 27 = 0

Use the discriminant to determine whether the quadratic equation
has two unequal real solutions, a repeated real solution, or no
real solution, without solving the equation.
5x^2−9x+6=0
Choose the sentence that describes the number of real solutions
to the quadratic equation.
A.There are two unequal real solutions.There are two unequal
real solutions
B.There is no real solution.There is no real solution.
C.There is a repeated solution.

x^5 +x^3 +x +1=0
use the IVT and Rolle's theorem to prove that the equation has
exactly one real solution.

1-classify the quatric in R^3 with the equation , then determine
the center of each.
-3 x2
+7y2 +72x +126y +z +95=0
2- prove that the quadric surface E with equation is a
hyperbolic paraboloid. Determine its center
y - yz =xz

Use the intermediate value theorem to show that there
is a root of the given equation in the specified interval
1. X^4 + x - 3= 0, (1,2)

Show complete solution
Use Green’s Theorem to determine the circulation curl
(tangential form) of ∮ 3???? + 2???? on the curve C , a rectangle
bounded by x = -2, x = 4, y = 1, and y = 2.

1.49 Two particular solutions of the equation
4f''(z)+f(z)=4f'(z) are f1(z)=ez/2 and
f2(z)=zez/2.
a) Show that any linear combination Af1(z) +
Bf2(z) is also a valid solution to the differential
equation.
b) Find the particular solution that satisfies the conditions
f(2)=14e and f'(2)=12e.

Determine the stability region using the Schur-Cohn criterion
for a system whose characteristic equation P(z).
P(z)=z^3 +0.5z^2 +kz-k=0

for the equation f(x) = e^x - cos(x) + 2x - 3
Use Intermediate Value Theorem to show there is at least one
solution.
Then use Mean Value Theorem to show there is at MOST one
solution

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