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Complex Analysis

Analytic Function Proofs and Real Functions

1) Show that the real part of the function z^(1/2) is always positive. 2) Suppose f: G --> C ( C complex plane) is analytic and that G is connected. Show that if f(z) is real for all z in G, then f is a constant.

Complex Eigenvalues Investigated

Find the solution to the given system for the given initial condition x'(t)= [1,0,-1;0,2,0;1,0,1]x(t) for a.) x(0)= [-2;2;-1] b.) x(-pi)=[0;1;1]

Continuity complex plane

Let G be an open subset of C ( complex plane) and let P be a polygon in G from a to b. Use the following 2 theorems to show that there is a polygon Q in G from a to b which is composed of line segments which are parallel to either the real or imaginary axes. The 2 theorems are: 1). Theorem: Suppose f: X --> omega is continuou

Metric spaces and the topology of complex plane

Show that { cis k : k is a non-negative ineger} is dense in T = { z in C ( C here is complex plane) : |z| = 1 }. For which values of theta is { cis ( k*theta) : K is a non-negative integer} dense in T ? P. S. cis k = cos k + i sin k, i here is square root of -1. I want a full justification for each step or claim.

Stereographic projection on complex plane

Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p / S = V ( / = intersection). Recall from analytuc geomerty that P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}. Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_3)

DeMoivre's Theorem and Power of Complex Numbers

See the attached file for complete equations --- Use DeMoivre's Theorem to find the indicated power of the following complex numbers: 1. Find the fourth roots of 256(1+3i ) 2. Find all solutions of the equation x3  27i = 0 [please show all the steps, including the algebraic ones] ---

Harmonic Functions and Cauchy-Riemann Equations

15. Determine whether or not the given function u is harmonic and, if So, in what region. If it is, find the most general conjugate function v and corresponding analytic function f (z). Express f in terms of z. Please see the attached file for the fully formatted problems.

Complex Variables : Region of Flow, Fluid Presure and Speed of Fluids

4. Show that the speed of the fluid at points on the cylindrical surface in Example 2, Sec. 108. (below) is 2A| sin θ| and also that the fluid pressure on the cylinder is greatest at the points z = ±1 and least at the points z = ± i --- - Show the speed of the fluids... - At an interior point of a region of flow...

Two Complex Variable Problems

See the attachment for the problems. --- - the image of the closed rectangular region... - principal branch of the square root... --- (See attached file for full problem description).

Four complex variable problems (87-4,5,6,10)

See the attachment for the problems --- - Find the bilinear transformation that maps... - Show the disposition of two linear fractional transformations... - A fixed point of transformation w = f(z)... - Show that there is only linear fractional transformation that maps... --- (See attached file for full problem descri

Four complex variable questions (83-5, 85-5, 7, 13)

See the attachment for the problems. --- - Find the region onto which the half plane y>0... - Find the image of the quantrant x>1, y>0... - Describe geometrically the transformation w=1/(z-1). - Using the exponential formz = re^(i x theta) of z... ---- (See attached file for full problem description)

Decision Analysis with Payoff Tables and States of Nature

A payoff table is given as States of Nature .... a. What decision alternative would be made using a minimax regret? b. What decision alternative would be made by using a conservative approach c. What decision alternative would be made by using a maximax approach?

Decision Analysis with Payoff Tables and States of Nature

A payoff table is given as ..... a. What choice should be made by the optimistic decision maker? b. What choice should be made by the conservative decision maker? c. What decision should be made under minimax regret? d. If the probabilities of E, F, and G are .2, .5, and .3, respectively, then e. What choice should be ma

Sum and difference formula

Please see the attached file. Solve: sin 285 = sin (225 + 60) [This is an easy problem, but I need to see all the steps that lead up to the following answer;] -2/4(3-1) [I got this answer: (-2 - 6)/4 I guess I'm really asking to see the steps in factoring my answer in order to match the abov

Diffusion Equations : Separation of Variables

4. (Separation) Seeking a solution u(x, t) = X(x)T(t) for the given PDE, carry out steps analogous to equations (3)?(6), and derive ODE's analogous to (7a,b). Take the separation constant to be ?K2, as we do in (6). Obtain general solutions of those ODE's (distinguishing any special ic values, as necessary) and use superposition

Simple Multiple Angle Formula Solutions using DeMoivre's Theorem

The double angle formulae are easy to learn: Sin 2x = 2 sin x cos x Cos 2x = (cos x)^2 - (sin x)^2 but working out Cos 4x say in terms of cos x and sin x by using identities such as Cos(A + B) = Cos A Cos B - Sin A Sin B is laborious. Find a simple method to work out any multiple angle formula, using De Moivre'