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Complex Analysis

DeMoivre's theorem

Suppose we want to find the nth root of a complex number z; that is, we want to find w such that We write z and w in the form Using DeMoivre's theorem we have (using the equation ) i) Show that the absolute value of wn and z are equal. Hence find R in terms of r. ii) Equate the arguments of wn and z to find a

Riemann sum and Continuity

Given f(x)=x^3 on [0,1] and the partition P={0,1/8,1/3,2/3,1}, find four different Riemann sums R(f,P). Show that Chi_Q is discontinuous at every point where Chi_Q is the characteristic function for Q - Rational numbers.

Quantitative Analysis: Networks, PERT and Critical Path

In one of the little-know battles of the Civil War, General Tecumseh Beuregard lost the Third Battle of Bull Run because his preparations were not complete when the enemy attacked. If the critical path method had been available, the general could have planned better. Suppose that the following planning network, with activity t

Gaussian elimination diagram

1. Solve the following system of equations by hand. Use the Gaussian elimination, on the augmented matrix, and write the row operation you used next to each new row. x + y + z = 0 3x - 2y + 2z = -14 2x + 3y - z = 22 2. Find all roots of the equation z^5 = i, i.e. find the five values of i^(1/5) and show them on an Argand

Solve the equation sinz=2 for z by

Solve the equation sinz=2 for z by: a) equating real parts and then imaginary parts in that equation b) using sin^-1(z)=-ilog[iz+(1-z^2)^1/2]

Pully equation with complex numbers (Cartesian/Polar)

Task 3 For the pulley shown below, the tension T in the cable is 100N. First express each tension in Cartesian form and add these together to find the resultand force on the pulley shaft of diameter 12mm. Then express the resultant force in Polar form. Hence determine the shear stress in the pulley shaft. given that it is in do

Derive the complex form of the Cauchy-Riemann equations.

Recall that if z=x+iy then, x=(z + zbar)/2 and y=(z-zbar)/2. a) By formally applying the chain rule in calculus to a function F(x,y) of two real variables, derive the expression dF/dz=(dF/dx)(dx/dz)+(dF/dy)(dy/dz)=1/2((dF/dx)+i(dF/dy)). b) Define the operator d/dz=1/2((d/dz)+i(d/dy)) suggested by part (a) to show that i

set of complex numbers

1.Find an example of a sequence an of complex numbers such that the series SUM a_n converges (conditionally), yet the series SUM a^(3)_n diverges. 2.Determine the set of complex numbers z for which the series SUM(1â?'z^2)^n converges. SUM means sigma.

Waiting line analysis

Student arrive at the Administrative Services Office at an average of one every 15 minutes, and their requests take on average 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eights hours per day. Assume Poisson arrivals and exponential service times. A. What percentage o

Computing Seasonal Index for Time Series Data

Air pollution control specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time series data exhibit seasonality, with the levels of pollutants showing similar patterns over the hours in the day. On July 15,16, and 17, the observed levels

Calculus: transformation of complex numbers

1.) Consider the transformation w = (i - z) / (i + z) . Show that the upper half plane Im z > 0 maps to the disk |w| < 1 and the boundary of the half plane maps to the boundary of the disk (the circle |w| = 1. 2.) Find the image of the semi-infinite strip x > 0, 0 < y < 1, under the transformation w = i/z. Sketch the s

Crashing Problems

Problem 1 - Crashing Development of a new deluxe version of a particular software product is being considered. The activities necessary for the completion of this are listed in the table below. Activity Normal Time Crash Time Normal Cost Crash Cost Immediate Predecessor A 4 3 2000 2600 - B 2 1 2200 2800 - C 3 3 500 50

Project Scheduling

1. The Mohawk Discount Store is designing a management training program for individuals at its corporate headquarters. The company wants to design the program so that trainees can complete it as quickly as possible. Important precedence relationships must be maintained between assignments or activities in the program. For e

Schwartz reflection principle

(A version of the Schwartz reflection principle.) Let the function f be continuous in the region { z : | z | < 1, Im z &#8805; 0}, real valued on the segement ( - 1, 1) of the real axis, and holomorphic in the open set { z : | z | < 1, Im z > 0}. Use the result: If f is a continuous complex-valued function in the open sub

Complex Valued Functions

Define the complex-valued function of a complex variable f: C --> C by f(z) = [IM(z)]^2 If S = {z is an element of C: Im(z) = 0} is the real axis, show using the definition of the derivative that f is differentiable at every z is an element of S. Show further that f is not differentiable elsewhere, ie. f is not differenti

Decision Analysis Theory

What is decision theory? What is the difference between a payoff table and an expected payoff table? In the following payoff table, let P(S1) = 0.30, P(S2) = 0.50, and P(S3) = 0.20. Compute the expected monetary value for each of the alternatives. What decision would you recommend? State of Nature Al

Singular Values Matrix

Let A be a (m x m) Normal Matrix a) Calculate the singular values of A as a function of the eigenvalutes of A. b) Prove that the eigenvalues of A are all purely imaginary if and only if A is anti-symmetric (A^H = -A)

Complex Analysis Problem (analytic and harmonic functions)

(a) Show that an analytic function f(z) defined in a simply connected domain Ω is constant if R(f(z)) (= the real part of f(z)) is constant throughout Ω. (b) Let f(z) be analytic and non-vanishing in a domain Ω. Show that ln l f(z) l is a harmonic function in Ω. Textbook: "Fundamentals of Complex Analysis with Appl

Decision Analysis and Stable Supply Surplus

Consider the following Pay-off Table: Decision Alternatives.........Shortage........Stable Supply........Surplus Motel...................................$-8000..............$15,000............$20,000 Food Restaurant.................$2,000................$8,000.............$6,000 Movie Theater.....................$6,000....

Decision Analysis

A decision maker who is considered to be a risk taker is faced with this set of probabilities and payoffs. State of Nature Decision s1 s2 s3 d1 5 10 20 d2 -25 0 50 d3 -50

Sum of Complex square roots of -16

I have a problem to work out, but I am having problems trying to come up with an answer. The problem is finding the sume of the complex square roots of -16. I think it starts out as sq rt of -16 = sq rt of 16i, so would that be 4i + 4i? If not, would you please help me to work out this problem? Thank-you.

Applying Sine Law

Please show work 18. Determine the remaining sides and angles of each triangle ABC. A = 18.75 degrees B = 51.53 degrees C = 2798 yd 20. Solve each triangle. C = 28.3 degrees b = 5.71 in a = 4.21 in 20. Find the sum of each pair of complex numbers. -5 -8i, -1 2. Find each power. [2(cos135degrees + i*sin135

Using discriminants

1. Determine the number of solutions and classify the type of solutions for each of the following equations. Justify your answer. a) x2 + 3x - 15 = 0 b) x2 + x + 4 = 0 c) x2 - 4x - 7 = 0 d) x2 - 8x + 16 = 0 e) 2x2 - 3x + 7 = 0 f) x2 - 4x - 77 = 0 g) 3x2 - 7x + 6 = 0 h) 4x2 + 16x + 16 = 0 2. Find an equatio

Calssify Numbers as Real/Rational or Real/Irrational

Can you please also explain how you come up with your answer after doing it. Thanks 1. Classify the given numbers as real and rational, real and irrational, or complex. Remember to simplify BEFORE you classify! a. (3)1/2 + 2i b. 3.14 c. &#960; d. (16)1/4 + 6 e. (18)1/3 f. 1/6 g. 6i h. 3 - (-9)1/2 Rational Nu

Explain why no negative values are chosen as values

If y=sqrtx-2, fill in the following table for x= 0, 1, 2, 3, 4. round to three decimal places where necessary. B) Explain why no negative values are chosen as values to substitute in for x. Please show the steps i try to solve it but do not get it.

Radical and Rational exponents

1) Solve algebraically. Trial and error is not an appropriate method of solution. You must show all your work. Learn how to type math roots and fractions by clicking on the link in the assignment list. Alternately, you may type as cuberoot(x) and show raising to the nth power as ^n, like x 3 is typed x^3. a) An

Complex number system

This number system is an extension of the real number system. Any polynomial equation has n complex roots, in general, in the complex number system. For example, the equation x2 + 1 = 0 has no real roots, but it has two complex roots given by +I and -I, where I is the "imaginary unit" given by the "square root" of -1. The com

Polynomials and Complex Roots and the Real Number System

This number system is an extension of the real number system. Any polynomial equation has n complex roots, in general, in the complex number system. For example, the equation x2 + 1 = 0 has no real roots, but it has two complex roots given by +I and -I, where I is the "imaginary unit" given by the "square root" of -1. The com

Bijective Conformal Mapping

A) find a bijective conformal mapping that takes a bounded region to an unbounded region b) prove that a conformal map cannot take a simply connected region onto a region that is not simply connected.