## Mathematical Methods in Physics and Engineering, 2004:

Office hours: M: 3-4, WF 11-12 or by appointment.

Office: APM 5256, tel. 534-2734

Course material: Fall term: It is planned to cover most of Chapters 1-5 from the book `Applied Complex Variables' by John W. Dettman. In particular, this includes complex differentiation, Cauchy-Riemann equations, Cauchy theorems, power series and analytic functions and residue theory (a useful tool for computing integrals). After that we will study some of the applications in the second part of the book such as potential theory, asymptotic expansions and/or differential equations (to be decided later in the term).

Computation of grade: Final: 60% Homeworks: 40%

Dates for homework to be turned in: 10/15, 11/5, 11/19, 12/3 or earlier. Get started early!

Homework assignments: (ask if some of the symbols do not make sense to you)

for 10/15: 1) Draw the regions D_1 = { z \in C, |z|<1, 0 < arg(z) <2 \pi /6 } and D_2 = { z^4, z\in D_1}

2) p. 12: 7, 8, p. 17: 6acd, p. 26: 2a !!OR!! 2c, look at problems 1 and 3 (solutions are given)

3) p 39: 5 do it both without using Cauchy-Riemann conditions and with using it, p 43: 5

4) Compute all possible values of (8i)^{1/3}, i^i, log[(1+i)^{\pi i}] and [log(1+i)]^{\pi i}.

5) Find complex numbers $z_1$ and $z_2$ such that $log(z_1z_2)\neq log(z_1)+ log(z_2)$. What are the possible numbers by which $log(z_1z_2)$ and $log(z_1)+ log(z_2)$ may differ?

for 11/5: p. 86: 2b, 3, p 101: 6, p. 112: 4, p. 119: 4, look at: p86: 1a, d, p 125: 5 (need not be turned in, but similar problems could appear in final

2(a) Show that the power series \sum z^n converges uniformly to the function 1/(1-z) on the region D_r={z, |z| less or equal r} for a fixed r<1.

(b) Does this series converge uniformly to 1/(1-z) on the region {z, |z|<1}? Does it converge pointwise?

for 11/19: p. 204: 1c,d, 2c p. 211: 3, 6 (look at Examples 5.2.1 and 5.2.3); do not use Jordan's Lemma.

Evaluate the integral of the function f(x)= x cos(kx)/(x^2-2x+10) over the whole real line, where k is real and positive. You will need Jordan's Lemma.

Show that all the roots of 9z^5+5z-3=0 lie in the annulus 1/2 < |z|<1 (Hint: look at Rouche's theorem, p. 213). How many roots are in each quadrant? Be careful: at least one root has to be real, as any polynomial of odd degree has a real root.

Do the following problem (need not be turned in): Compute the integral from 1 to infinity of the function x/(x^2+4)\sqrt{x^2-1}, where \sqrt{y} means the positive square root of the positive number y.

for 12/3 or earlier (recommended, so that you can get a feedback by Friday): 1(a) Show that if the Moebius transformation f(z)=(az+b)/(cz+d) maps the real line onto itself then one can assume all scalars a,b,c,d to be real.

(b) Find all Moebius transformations which map the upper half plane {z, Im(z)>0} to itself.

(c) Find a Moebius transformation which maps the real line to the unit circle.

(d) Do exercise 2 on p. 51 (Hint: If s and t are Moebius transformations as in in part (b) and in part (c), consider transformations of the form tst^{-1}).

2. Let u(x,y) be a solution of the Dirichlet problem on the unit disk with boundary condition u(1,t)=g(t), where (r,t) are polar coordinates. Show that the value of u at the origin (0,0) is equal to the average of g(t), 0 < t < 2\pi .

3. Consider two circles of radius 1 and centers 1 and i respectively. Let D be the intersection of the corresponding disks. Solve the Dirichlet problem for this region, with boundary conditions g(z)=1 on the upper arc, and g(z)=0 on the lower arc. (Hint: Use a Moebius transformation which maps the two circles to two straight lines [this forces one of the intersecion points to be mapped to infinity; make a convenient choice for one of the lines]; then solve the problem for the resulting region).

Look at (need not be turned in; please note change in sign!): Find the first three nonzero terms in the asymptotic expansion in x of the integral in t from 0 to infinity of the function exp(-t^4 - xt^2).