Expand the function f(z) = (z − 1) / z^ 2 (z + 1)(z − 3) as a Laurent series about the origin z = 0 in all annular regions whose boundaries are the circles containing the singularities of this function.

Let S = {-3, -2, -1, 0, 1, 2, 3}. Define a relation R on S by: xRy if and only if x = y + 4n for some integer n.

a) Prove that R is an equivalence relation.

b) Find all the distinct equivalence classes of R.

Let T: V →W be a linear transformation from V to W.

a) show that if T is injective and S is a linearly independent set of vectors in V, then T（S） is linearly independent.

b) Show that if T is surjective and S spans V,then T（S） spans W.

Please do clear handwriting!

What are the Aspects, Application, and Solutions for a Stochastic Differential Equation?

A. Find y as a function of x if

y′′′ − 13y′′ + 40y′ = 0,

y(0) = 2, y′(0) = 9,  y′′(0) = 1.
y(x) =

B. Find y as a function of x if

y⁽⁴⁾ − 10y′′′ + 25y′′ = 0,

y(0) = 11, y′(0) = 13, y′′(0) = 25, y′′′(0)=0.
y(x) =

C. Find y as a function of x if

y′′′ − 4y′′ − y′ + 4y = 0,

y(0) = −6, y′(0) = −5, y′′(0) = 24.
y(x) =

Complex Analysis

60. Show that if f(z) is analytic and f(z) ≠ 0 in a simply connected domain Ω, then a single valued analytic branch of log f(z) can be defined in Ω

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x,y) ∈ R if and only if:

a) x = 1 OR y = 1

b) x = 1 I was curious about how those two compare.

I have the solutions for part a) already.

Define phase term (fourier transform)

2. Assume that the sum is fixed at the point (0,0) in the x,y plane. The path of a comet around the un is given by the equation y = x² - 0.5 in astronomical units. (One astronomical unit is the distance between the sun and the Earth).

a. Use a graphing tool, such as Desmos, to graph the function.

*I have done the part and got the graph*

b. Find the coordinate of the point where the comet is closest to the sun. (Recall that the distance of a coordinate from the origin is given by Pythagoras theorem d = sqrt(x² + y²).

c. If the Earth is at the point (1,0) when the comet passes by, what are the coordinates of the point where the comet is closest to the Earth? (Recall: distance between (x,y) and (x₁,y₁) is d = sqrt((x-x₁)²+(y-y₁)2

Thank you so much and stay safe!

Assignment problem and branch and bound

A factory produces a certain type of car parts. There are four alternative machines that can be used for the production of the car parts from start to finish. Each of the machines needs to be controlled by an individual operator. The operators have different efficiencies on different machines. The table below shows how many car parts the individual operators produce in average per day. Furthermore, this table shows how many erroneous parts the individual operators produce in average. Your task is to find out where the operators should be placed such that they produce as many as possible car parts. At the same time, the number of erroneous parts should not exceed 4 % of the total production.

Production per day:

Number of erroneous parts per day:

a) Set up a mathematical program for this problem.

Find the Laplace Transform of the functions

t , 0 ≤ t < 1

(a) f(x) =     2 − t , 1 ≤ t < 2

0 , t ≥ 2

(b) f(t) = 12 + 2 cos(5t) + t cos(5t)

(c) f(t) = t 2 e 2t + t 2 sin(2t)

6.(a) Show that if f : [a,b]→R is Riemann integrable and if m ≤ f (t) ≤ M holds
for all t in the subinterval [c,d] of [a,b], then
m(d −c) ≤ ∫_c^d f(t) dt ≤ M(d −c). (that is supposed to be f integrated from c to d)

(b) Prove the fundamental theorem of calculus, in the form given in the Introduction
to this book. (Hint: Use part (a) to estimate F(x)−F(x₀)/x−x₀.)

2
Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and w=(3,5,4,4).

2.1
Construct a basis for the vector space spanned by u, v and w.

2.2
Show that c=(1,3,2,1) is not in the vector space spanned by the above vectors u,v and w.

2.3
Show that d=(4,9,17,-11) is in the vector space spanned by the above vectors u,v and w, by expressing d as a linear combination of u,v and w.

Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0 Derive the general solution of the given differential equation about x = 0. Your answer should include a general formula for the coefficents.