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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?

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)


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

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)

Complex Variables : Limits

11. T (z) = (az + b) / (cz + d) (ad - bc &#8800; 0) Show the following. (Please explain by using theorem.) (a) lim T (z) = &#8734; if c = 0 z&#8594;&#8734; (b) lim T (z) = a / c and lim T (z) = &#8734; if c &#8800; 0. z&#8594;&#8734; z&#8594;-d/c

Complex Variables : Limits

10. Show the following limits. (Please explain by using theorems.) (a) lim 4z2 / (z - 1)2 = 4 z&#8594;&#8734; (b) lim 1 / (z - 1)3 = &#8734; z&#8594;1 (c) lim (z2 + 1) / (z - 1) = &#8734; z&#8594;&#8734;

Comples Variables : Limits

5. Show that the limit of the function _ f (z) = ( z / z )2 as z tends to 0 does not exist. Do this by letting nonzero points z = (x, 0) and z = (x, x) approach the origin. (Note that it is not sufficient to simply consider points z = (x, 0) and

Simplify a Complex Expression

Suppose that f (z) = x2 - y2 - 2y + i&#1468;(2x - 2xy), where z = x + iy. Use the expressions _ _ x = (z + z) /2 and y = (z - z)/2i to write f (z) in terms of z, and simplify the result.

Complex Variables: Verify Inequality

3. Verify that (sqrt(2))&#1468;|z| &#8805; |Re z| + |Im z|. Suggestion: Reduce this inequality to (|x| - |y|)2 &#8805; 0. Sqrt(2) means square root of 2.

Kirchoff's Laws : Mass-Spring Equation

Consider a basic electric circuit with a resistor, capacitor, and inductor and input voltage V(t). It follows Kirchoff's Laws that the charge on the capacitor Q = Q(t) solves the differential equation: {see attachment}, where L (inductance), R (resistance), and C (capacitance) are positive constants (depending on material). The