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Complex Analysis

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)

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 z≠ 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

11. T (z) = (az + b) / (cz + d) (ad - bc ≠ 0) Show the following. (Please explain by using theorem.) (a) lim T (z) = ∞ if c = 0 z→∞ (b) lim T (z) = a / c and lim T (z) = ∞ if c ≠ 0. z→∞ z→-d/c

Complex Variables : Limits

10. Show the following limits. (Please explain by using theorems.) (a) lim 4z2 / (z - 1)2 = 4 z→∞ (b) lim 1 / (z - 1)3 = ∞ z→1 (c) lim (z2 + 1) / (z - 1) = ∞ z→∞