Numbered exercises refer to the
Lecture notes on Complex Calculus that you have been studying. There are no problems to hand in. You should work on the problems listed below in preparation for exams.
Homework 25, Mon. 4/5/2010 : Study sections 1.1, 1.2, 1.3 of the complex calc notes. What is a complex number? modulus? complex conjugate? addition, subtraction, multiplication, division. Geometric sum for arbitrary complex number.
Exercises: 1.3. 1, 2. Does formula (9) work for all q's? Does formula (10) work for all q's?
Prove formula (8) in the notes using geometric sum.
Homework 26, Wed. 4/7/2010 : Study sections 1.1--5, of the complex calc notes. Geometric series, ratio test, applications and connections with Taylor series
Plot 1/(1-q) and 1+q+...+qN for real q's in the interval [-5,5] for a few N's (say, N=1,2,3,4,5,6). What do you observe?
Ratio test: nice recipe, but why does it imply convergence? What if the limit of the ratio is greater than 1?
What is a power series? What is a Taylor series?
Exercises: 1.5. 1--7
Homework 27, Fri. 4/9/2010 : Study sections 1.1--6, of the complex calc notes. Taylor series. TS for ln(z), exp(z), cos(z), sin(z)
Where in the z-plane does a power series converge? What is the `radius of convergence'?
Consider the TS of 1/(1+z2) about z=3. What is its radius of convergence?
(does `consider' mean `calculate'?)
Must know TS for exp(z), cos(z), sin(z)
Derive (prove) the formula (26), (31), (32), (33), starting from the series definition of the functions of course.
Exercises: 1.6. 1, 2, 3, 4, 5, 6
Homework 28, Mon. 4/12/2010 : Study sections 1.1--8, of the complex calc notes. z=r ei θ = |z| ei arg(z), θ=arg(z) + 2 k π, ln z,...
what is arg(z)?
what is ln z? why?
What is ln(-8)?
Find all z's s.t. ez = -8.
What is iii?
You are responsible for 1.8.3: roots and the fundamental theorem of algebra.
Homework 29, Wed. 4/14/2010 : Study sections 2.1, of the complex calc notes. f(z)=u(x,y) + i v(x,y), df/dz ≡ f '(z), Cauchy-Riemann, Laplace's equation, harmonic functions
Must know Cauchy-Riemann equations, what they mean and where they come from.
Show that z* is continuous. Show that z* is not z-differentiable.
Exercises: 2.1. 1, 2
Find a non-zero solution φ(x,y) to Laplace's equation in the polar wedge 0 < θ <π that is zero on the radials θ=0 and θ=π.
For 2D ideal fluid flow (i.e. irrotational and incompressible) φ(x,y) would be a streamfunction, its contour lines would be the streamlines.
The fluid velocity is (∂φ/∂y, - ∂φ/∂x), so the speed is inversely proportional to the distance between contours (you should be able to explain this latter statement).
For 2D electrostatics, φ(x,y) would be an electric potential, its contours would be the equipotentials,
its gradient would be the electric field (with a minus sign).
Show that the imaginary part of z2 provides a non-zero solution φ(x,y) to Laplace's equation in the polar wedge 0 < θ <π/2 that is zero on the radials θ=0 and θ=π/2. Plot contours of φ(x,y) (you should not need help from a software to do that).
Find a non-zero solution φ(x,y) to Laplace's equation in the polar wedge 0 < θ <π/4 that is zero on the radials θ=0 and θ=π/4. Plot contours of φ(x,y) (you may use software to help you).
Show that the imaginary part of z1/2 provides a non-zero solution to Laplace's equation in the polar wedge 0 < θ < 2 π that is zero on the radials θ=0 and θ=2π. How is z1/2 defined?!
Plot contours of φ(x,y) (you may use software to help you).
Explain why the derivative of Matlab's z1/2 does not exist along the negative real axis.
Homework 30, Fri. 4/16/2010 : Study sections 2.1, 2.2 of the complex calc notes. Visualizations of f(z), preservation of angles, conformal mapping
Homework 31, Mon. 4/19/2010 : Study sections 2.1, 2.2 of the complex calc notes. preservation of angles, conformal mapping
If a function f(z) is differentiable at z=x+iy, and f '(z) ≠ 0, what is the angle between the isocurves u(x,y)=u0 and v(x,y)=v0 at the point (x,y)? Why?
What happens if f '(z)=0?
When and why do complex mappings w=f(z) preserve angles? What does that even mean?!
What happens to angles under the maps (i) w=z2, (ii) w=z3, (iii) w = (z-1)2, (iv) w= sqrt(z) ?
Derive and understand all the pretty pictures, much of it is application of pre-calculus (parabolas, hyperbolas, ellipses,...) and math 222 (implicit and parametric curves) applied to complex functions.
Homework 32, Wed. 4/21/2010 : Study sections 3.1 of the complex calc notes. complex integral, relation to line integrals
Compute the integral of z2 from z=i to z=3i over the half circle joining i to 3i (a) clockwise,
(b) counterclockwise, (c) over the straight line from i to 3i.
What would be the fundamental theorem of Calculus for the integral of a complex function f(z) over a curve C in the complex z plane?
How would you justify (or prove) it?
We computed the integrals of z2, 1/z and sqrt(z) over the unit circle in class.
What results do you obtain if you use the fundamental theorem of calculus to compute those integrals?
Calculate the integral of 1/z from -1+i to -1-i along the counterclockwise circular arc centered at 0 by explicit parametrization and by the fundamental theorem of calculus. Do your answers match? why/why not?
Write (63) explicitly for f(z)=z2 and compute the imaginary part of that integral for the path C that consists of the straight lines from z=i to z=1+i to z=2+3i. (Note that this path is not closed)
Remember Stokes theorem? What is Stokes theorem? What does Stokes theorem look like for a vector field
v = F ex + G ey (where F=F(x,y) and G=G(x,y) of course) and a closed curve C in the (x,y) plane?
Homework 33, Fri. 4/23/2010 : Study sections 3.1 of the complex calc notes. Cauchy's theorem, applications
What is `Cauchy's theorem'?
How does one prove Cauchy's theorem?
What is the connection, if any, between complex integrals and work done by forces in physics/mechanics?
How does one apply Cauchy's theorem to the integral over a closed loop of f(z)=1/(z2+1)? What are ALL the possible values of such integral as a function of the closed contour C? (arbitrary but non-self intersecting closed contour)
Derive and digest formula (67).
Exercises: 3.1.1--9.
Homework 34, Mon. 4/26/2010 : Study sections 3.1, 3.2 of the complex calc notes. `Calculus of Residues', poles/singularities, Cauchy's formula, generalized Cauchy's formula.
Derive and digest formula (67) (yes, you already did this in the previous homework, but it is a key result).
Derive and digest formula (69) (Cauchy's formula) (1) as done in the notes, (2) using Taylor series and result (67).
Derive and digest formula (79) (Generalized Cauchy formula) using Taylor series and result (67).
What are all the possible values of the integral over a closed loop of f(z)=(z+1)-1exp(1/z)? (done in class)
What are all the possible values of the integral over a closed loop of f(z)=z2 sin(1/z)?
What are all the possible values of the integral over a closed loop of f(z)=(z+1)-1z-2?
Review Exercises: 3.1.1--9.
Exercises: 3.2. 7
Synthesize. What are the key ideas/steps involved? How do you justify these key ideas/steps?
Homework 35, Fri. 4/30/2010 : Study sections 3, 4 of the complex calc notes. Applications of complex integration, a.k.a. contour integration, `1st and 2nd Key Examples' in the notes + exercises 1, 2
Homework 36, Mon. 5/3/2010 : Study section 4 of the complex calc notes. Applications of complex integration, a.k.a. contour integration: Example 3 , also try out eqn (87) and the integral from -∞ to +∞ of (cos x)/(1+x2)
Homework 37, Wed. 5/5/2010 :
Example 4 in section 4 of complex calc notes,
lots of subtleties in that classic problem!!