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# Complex Analysis

### Complex Analysis : Complex Potential for a Fluid, Velocity and Stream Functions

A. A certain fluid is flowing with a constant speed V in a direction making an angle B with the positive x-axis. Find the complex potential for the fluid under consideration. Also determine the velocity and the stream functions.

### Show that the derivative of the following function does not exists at any point f(z) = 2x + ixy^2 Note that I and j are used and they represent the same imaginary quantity ( - 1)^(1/2).

Complex Variables Existence of a Function Show that the derivative of the following function does not exists at any point

### Proof of Trignometric Relationship Using Complex Numbers

Given that cos x=(e^jx + e^-jx)/2 and sin x=(e^jx - e^-jx)/2j Using ONLY this information, prove that: cosAsinB=1/2[sin(A+B)- sin(A-B)]. See the attached file.

### Correlation Coefficient : Correlating Data

Using the following data, obtained from an imaginary harness small manufacturing company, determine the correlation between the number of hours that an employee works and how many belts they produce: Hours Worked 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Belts Produced 2 3 3 4 5 6 6

### Solve the Complex Number Equality

Please show me how to solve the following : x^4 = 5 - 5i

### Complex Conjugates

What is the complex conjugate of 78.93iw^3+30.48iw^2+iw? How did you find it?

### Weighted Voting System

1) For each weighted voting system , find all dictators (d), veto power players( vp), and dummies (d) ( 7: 7,3,2,1) 2) For the weighted voting system ( 12: 6,4,3,1,1,) a) Find what percent of the total vote is the quota. b) In a Shapley-Shubik distribution system , how many sequential coalitions would be formed from thi

### Conformal Mapping

Find a conformal mapping from the unit disc &#916;(0,1) = {z|z|<1} to D={z:|z|<1}[0,1]. Please see attached for diagram.

### Cauchy's Formula

Use Cauchy's formula for the derivative to prove that if f is entire and |f(z)|&#8804; A|z|Â² + B|z| + C for all z&#949;C, then f(z) = azÂ² +bz + c Please see attached for full question.

### Simplify the Complex Number

Simplify the following (3+i )/2 + (1- i )/4.

### Complex Variable Class - Undergraduate 500 Level

Use the formal method, involving an infinite series of residues and illustrated in the examples. 7. F(s) = 1/(s cosh (s^Â½)) Please see attachment for full question.

### Complex Variable Class - Undergraduate 500 Level

The beta function is this function of two real variables... Make the substitution t=1/(x+1) and use the result obtained in the example in Sec. 77 to show that... Please see attachment for equations.

### Residues and Poles; Polar Numbers; Demoivre's Theorem

Please see the attachment for questions relating to residues and poles (polar numbers and Demoivre's Theorem). These problems are from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution.

### Complex Variables : Taylor Series

Problem: Show that when ... Please see the attached file for the fully formatted problems.

### Complex Variables : Taylor Expasions and Intervals of Convergence

Derive the expansions. Please see the attached file for the fully formatted problems.

### Complex Variables : Prove Equalities Using Taylor Expansions

Show that when z does not equal 0, a) e^2/z^2 = 1/z^2 + 1/z + 1/2! + z/3! + z^2/4! + ... (See attachment for other question)

### Complex Variables : Taylor Series Representation

Derive the Taylor Series representation ... (see attached)

### Complex Variables : Taylor Series

This problems is from complex variable class. Please specify the terms that you use if necessary and clearly explain each step of your solution. Problem: Obtain the Taylor series ... (see attachment) for the function ... (see attachment)

### Complex Variable : Complex Summation Proof

Prove that ... (see attachment for equation).

### Complex Variables : Prove the Complex Summation Identity

Write z=re^i0, where 0<r<1, in the summation formula that was derived in the example in Sec. 52. Then, with the aid of the theorem in Sec. 52 show that... (See attachment for details)

### Irreducibility Multiple Roots

Prove that f(x) has no multiple root over C ... (PLEASE SEE ATTACHMENT FOR FULL PROBLEM)

### Abstract Analysis : Continuity of a Map

Show that phi is (infinite d,d) continuous where d is the standard metric... (See attachment for full question)

### Boundary of a Square : Analysis with Trigonometric and Hyperbolic Functions

Let CN denote the boundary of the square formed by the lines... Please see the attached file for the fully formatted problems.

### Integrals of Complex-Valued Functions

Please see the attached file for the fully formatted problems.

### Complex Exponentials

Please see the attached file for full problem description. --- 5. Write |exp (2z + i) | and | exp (iz2) | in terms of x and y. Then show that | exp (2z + i) + exp (iz2) | &#8804; e2x + e-2xy. (exp means exponential function)

### Determine the singular points of the function

In each case, determine the singular points of the function and state why the function is analytic everywhere except at those points: (a) f (z) = (2z + 1) / [z (z^2 + 1)] (b) f (z) = (z^3 + i) / (z^2 - 3^z + 2) (c) f (z) = (z^2 + 1) / [(z + 2)(z^2 + 2z + 2)]

### Complex Variables : Differentiability

9. Let f denote the function whose values are _ f (z) = z^2 / z when z &#8800; 0, f (z) = 0 when z = 0. Show that if z = 0, then &#8710;w/&#8710;z = 1 at each nonzero point on the real and imaginary axes in the &#8710;z, or &#8710;x&#8710;y, plane. Then show that &#8710;w/&#871

### Complex Variables: Differentiation - Quotient Formula

7. Prove that dzn-1/ dz = nzn-1, for the derivative of zn remains valid when n is a negative integer (n = -1, -2, ???), provided that zâ‰  0. (Suggestion: Write m = -n and use the formula for the derivative of a quotient of two functions). See the attached file.

### Complex Variables : Limits and Differentiability

3. Give a direct proof that f Î„(z) = -1 / z2 when f (z) = 1 / z (z â‰  0). Use this definition to prove the problem. dw / dz = lim âˆ†w / âˆ†z âˆ†zâ†’0

### Complex Variables : Limits

11. T (z) = (az + b) / (cz + d) (ad - bc â‰  0) Show the following. (Please explain by using theorem.) (a) lim T (z) = âˆž if c = 0 zâ†’âˆž (b) lim T (z) = a / c and lim T (z) = âˆž if c â‰  0. zâ†’âˆž zâ†’-d/c