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Complex Analysis

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'

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

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|² + B|z| + C for all z&#949;C, then f(z) = az² +bz + c Please see attached 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

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 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)