Homework 1

Due: April 9, 2015 at end of workday

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1. Chapter 2, 5-66

Find x and y as functions of t for the example above, and verify for this case that v and a are correctly given by the method of the example. In the example, z = x + iy = (i+2t)/(t-i), and v = 3/(t²+1) and a = 6/(t² +1)^3/2

2. Chapter 2, 5-67

Find v and a if z = (1-it)/(2t+i)

3. Chapter 2, 6-1

Prove that an absolutely convergent series of complex numbers converges. This means to prove that the sum of (a_n+ib_n) converges if the sum of
(a_n²+b_n²) ^1/2 converges.

4. Chapter 2, 6-5

Test the sum (1/n²+i/n) for convergence

5. An infinite, grounded, conducting cylinder (radius a = 1) is placed in an electric field uniform at infinity. Assume that the axis of the cylinder lies along the z-axis and that the electric field at infinity is |E| = E_o in the y direction. The boundary conditions are that the electric potential V = 0 for r = a (the surface of the conducting cylinder) and V = - E_oy at infinity. Using the map

F(z) = z + 1/z

find the electrostic potential V everywhere. Here, z is not the coordinate but, rather, is z = x + iy. Sketch the equipotential surfaces and field lines.

6. Chapter 14, 11-35

Find expressions for the Cauchy-Riemann conditions in a general orthogonal curvilinear coordinte system.

7. Chapter 14, 2-24

Find out whether (y-ix)/(x²+ y²) is analytic.

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