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Complex Analysis

An application of Cauchy's inequality

Let f be an entire function such that |f(z)|<=A|z|. Use Cauchy's inequality to show that f(z)=az for some complex constant a. See the attachment for a more complete description of the question and Cauchy's inequality.

Some simple applications of the Cauchy-Goursat theorem

Use the Cauchy theorem to show that the integral around the unit circle |z|=1, traversed in either direction, is zero for each of the following functions: 1) f(z)=z exp(-z) 2) f(z)=tan(z) 3) f(z)=Log(z+2) The attached file contains this question written more clearly with correct mathematical notation.

Green's Theorem: Contour Integrals

** Please see attached file for the complete problem description ** Complex Analysis Problem Problem. Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by C can be written ..... Suggestion: You can use the form: R is a closed region for real valued functions. Does thi

Dividing by Zero and Imaginary Numbers

Mathematicians say that division by zero is forbidden. The expression 5/0, for example, is undefined. "Undefined" in this sense means "unable to be determined". Why is this? When we divide 5 by 5 (5/5) we get 1. Divide 5 by 2 we get 2.5. Each time we make the denominator smaller in the expression 5/x, the expression gets

Determine if the following equations are real or complex.

Determine if the following equations are real or complex; explain the answer in detail. Determine whether the following equations real or complex solutions; justify your answer. Note: It is not necessary to find the solutions; just determine if they are real or complex and explain why. a) 5x2 + 8x + 7 = 0 b) (7)1/2y2 - 6

Are the roots real, repeated real, or complex?

Determine if the following have a solution or not? justify answer. (apply the discriminant) are the roots real, repeated real, or complex? 1) 5x^2+8x+7=0 2) (7)^1/2y^2-6y-13(7)^1/2=0 3) 2x^2=x-1=0 4) 4/3x^2-2x+3/4=0 5) 2x^2+5x+5=0 6) p^2-4p+4=0 7) m^2=m+1=0 8) 3z^2+z-1=0

Solve the triangle and complex number

Please see attached file for full problem description. 1. B = 54 degrees, C = 112, and b = 18 2. Solve the equation on the interval [0, 2pi]: (cosx)^2 + 2 cos x + 1 = 0 3. Solve the triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. C = 110°, a = 5, b = 11 4. Solve the triangle.

Quantitative Analysis - Waiting lines and queuing Theory models

I have some Quantitative Analysis questions I need help understanding. Waiting lines and queuing Theory models 1. The New Providence shopping mall is considering setting up an information desk manned by one employee. Because of the complex design of the mall, it is expected that people will arrive at the desk at about twi

Complex Metric Spaces

Let (S,d) be a metric space and define the function u(x,y) = d(x,y)/(1+d(x,y)) for all x,y in S. (a) Prove that u is a metric on S with sup u(x,y) <= 1. (b) If S = C (complex) and d is the usual Euclidean metric d(z,w) = abs(z-w), then prove that sup u(z,w) = 1. (c) For 0 < r < 1, show that u(x,y) < r if and only

Complex Variables, Laurent Series and Uniform Convergence

(1) Let G = {z : 0 < abs(z) < R} for some R > 0 and let f be analytic on the punctured disk G with Laurent Series f(z) = sum a_n*z^n (from n = -oo to oo). (a) If f_n(z) = sum a_k*z^k (from k =-oo to n), then prove that f_n converges pointwise f in C(G,C) (all continuous functions from G to C (complex)); i.e., {f_n}

Derivation of Poisson Integral Formula for the Half-Plane

If is analytic in a domain containing the x-axis and the upper half-plane and in this domain, then the values of the harmonic function in the upper half-plane are given in terms of its values on the x-axis by (y>0). Below is an outline for the derivation, I just need to figure out how to justify the steps.

Complex Proof : Analytic Functions

Let f be an entire function such that |f(z)| &#8804; A|z| for all z in C for some fixed positive real number A. Use the attached theorem to show that f(z) = mz for some complex number m.

Quantitative Analysis in Business Decisions

Mr. Davison has been operating a small bicycle shop at the same location near Aspen, Colorado for 50 years. What type of decisions must he make in operating his business? On what basis would he likely be making these decisions? How do you think Mr. Davison would respond to a suggestion that he hire a quantitative analyst to assi

Complex Variables : Analytic Functions and Limits

Please help with the following mathematics-related problem. Let f(z) be analytic in a region G and setphi(z,w) = (f(w)-f(z))/(w-z) for w,z E G w does not equal z. Let z0 Ye G. Show that lim (z,w)-->(z0,z0) phi(z,w) =f'(z0). Complex Variables. See attached file for full problem description

Complex Variables: Rectifiable Path

Please help with the following problem. Fix w=re^i(theta)(not equal)0 and let gamma be a rectafiable path C-{0} from 1 to w. Show there is an integer k such that (integral)gamma z^-1 dz = log r + (theta)+ 2 pi i k See attached file for full problem description.

Fixed and Variable Cost Analysis

Costs can be classified into two categories, fixed and variable costs. These costs behave differently based on the level of sales volumes. Suppose we are running a restaurant and have identified certain costs along with the number of annual units sold of 1000. Item: Raw Materials (cost for hamburgers) Total Annual Cost: 650

Stereographic Projection

1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1). a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1 ps. here z bar means conjugate of z b) Show that the triangles Nz'z and NZZ' are similar. The

Holomorphisms

Suppose that f is holomorphic in a region G(i.e. an open connected set). How can I prove that in any of the following cases a)R(f) is constant b)I(f) is constant c)|f| is constant d) arg(f) is constant we can conclude that f is constant. Ps. here R(f) and I(f) are the real and imaginary parts of f.

Logarithm and Complex Numbers

See the attached file. 1. Given that s = 1.59t(1-3v), obtain the value of v when s = 3.52 and t = 21.56. 2. Solve log(2x + 3) = log(4x) + 2, for x giving the answer correct to 3 significant figures. 3. For a thermodynamic process involving a perfect gas, the initial and final temperatures are related by:

In the real world, where might these so-called imaginary numbers be used?

My question which I am not exactly sure is : 1.I need to find the mass and radius of three of the nine planets in our solar system. I need to be sure that the masses are expressed in kilograms and the radii are expressed in meters. 2.Using data, I need to find out how to calculate the gravitational acceleratio

Algebra: Complex Roots and Solutions, De Moivre's Theorem and Argand Diagrams

1 Find the real and complex solutions of these cubic equations. a) (z-3)(z2-5z+8)=0 b) z3 - 10z2- 34z- 40 = 0, given that 3-i is a root (solution). 2 Solve the equation z3 = 125 cis 45 3 Consider the complex number: z = = cos + z sin a) Use De Moivre's theorem to find z2, z4 and z6. Leave your answers in polar form. b) Pl