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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

### Show that transformation W (Z) = (a Z + b) / (c Z + d) of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition a d > b c

Show that transformation W (Z) = (a Z + b) / (c Z + d) of the upper half of a complex plane is 1-1 and onto the upper half plane if a, b, c, and d are real and satisfy condition a d > b c

### Dense Subsets of the Unit Circle in the Complex Plane

To find theta, and prove that {e^{in theta}: n non-negative integer} is a dense subset of the unit circle.

### 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)

### Complex numbers

What is the solution (-1+i squareroot of 3)exponent 9?

### 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+&#61654;3i ) 2. Find all solutions of the equation x3 &#61485; 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 Number Functions : Analyticity and Differentiability

Given f(z), determine f'(z), where it exists and state where f is analytic and where it is not: x + i sin y

### Complex Numbers: Polar and Cartesian Form

Obtain, in polar form all values z^(2/3), z^(3/2) and z^pi for each given z. Obtain, in Cartesian form all values z^i, z^(1-i) 2 + 2i. See the attached file.

### Point Sets of Complex Inequalities

With a labeled sketch, show the point sets defined by the following: |z|<|z-4| 2â‰¤|z+i|â‰¤5. See the attached file.

### Evaluate a Complex Expression

Evaluate |(2-i)^3 / (1+3i)^2|. See the attached file.

### 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 &#952;| and also that the fluid pressure on the cylinder is greatest at the points z = Â±1 and least at the points z = Â± i --- - Show the speed of the fluids... - At an interior point of a region of flow...

### Complex Variables : Stream Functions and Flow Around a Corner

(See attached file for full problem description) --- - flow around a corner... - components of velocity... --- (See attached file for full problem description)

### Analytic Functions, Contours, Polynomials and Rouche's Theorem

Please see the attached file for the fully formatted problems. --- - Suppose the function f is analytic inside... - Determine the number of zeros... - Write f(z) - z^n and... - Any polynomial... --- (See attached file for full problem description)

### Complex Variables : Analytic Funcitons and Contours

Please see the attached file for the fully formatted problems. --- - Lef f be a function which is analytic inside... - Write and equation... --- (See attached file for full problem description)

### 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 285ï‚° = sin (225ï‚° + 60ï‚°) [This is an easy problem, but I need to see all the steps that lead up to the following answer;] -ïƒ–2/4(ïƒ–3-1) [I got this answer: (-ïƒ–2 - ïƒ–6)/4 I guess I'm really asking to see the steps in factoring my answer in order to match the abov

### Complex Variables : Substitution and Beta Function

See the attachment (77-7) for the problem. and you should also see the examples in section 77 in order to solve it. I also attached section 77 (example1-1 and example1-2).

### Iterative Methods for Linear Systems : Euclidean Norm

The problem is from Numerical Methods. Please show each step of your solution and tell me the theorems, definitions, etc. if you use any. Thank you. (Complete problem found in attachment)

### 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

### Three-digit rounding arithmetic

Use three-digit rounding arithmetic to compute the following sums (sum in the given order): (See attachment for full question)

### 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'

### Complex Number : Write in standard form.

Write the expression i^6-5 in the standard form a + bi