Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner, making use of any relevant techniques discussed in the lectures or readings. This problem set has 375 points possible, and will be graded out of 100 points. 1. (15 points) (a) (5 points) Describe for what values of z the sequence of functions { [ ( Log z 1 z )]} n converges. (b) (10 points) Over what kinds of regions is the convergence uniform? For full credit, the regions you describe should be as large as possible. Justify your results rigorously. 2. (20 points) Over what regions in the complex plane do the following series converge uniformly? Prove your statements. For full credit, establish uniform convergence on regions as large as possible. (a) (10 points) (b) (10 points) n z 3 +n 3 ( ) n z + 1 z 2 3. (10 points) Consider the sequence of functions {f n (z)} with f n (z) = (z (1 1/n)) 1 1
and let C = {z C : z = e iθ, 0 θ < 2π} be the positively oriented unit circle. Calculate the following limits and integrals: lim n C f n(z) dz, C lim n f n (z) dz. Explain, using general mathematical analysis principles, why these quantities are or are not equal. Be sure to connect the specific functions explicitly to the analysis principles you are discussing, and justify all statements. 4. (10 points) Calculate the integral C dz P (z) where P (z) is a polynomial with degree greater than or equal to two, and C = {z : z = Re iθ, 0 θ < 2π} is a circular contour with positive orientation and radius R large enough that all the zeroes of P (z) are contained in the interior of C. Express your answer, as concisely as possible, in terms of properties of the polynomial P. 5. (5 points) Find the radius of convergence of the power series: (n + 2) n z n. n! n=0 6. (20 points) Find the largest domain in the complex plane over which the representation 1 f(z) = n z2 1 defines an analytic function. You do not need to attempt to analytically continue this representation, but do give a precise argument for why the representation of f(z) must be analytic over the domain you state. 7. (20 points) Consider the function defined by the following series: with f(z) = n= a n = 1 2πi C R a n z n Log z dz (1) zn+1 where C R is a positive oriented circle of radius R centered at the origin. This is how a Laurent series for Log z about its singularity at z = 0 would be 2
constructed except the theorems about the validity of Laurent series don t apply here. Compute this Laurent series for Log z anyway, and see whether it converges to Log z. (Feel free to explore this convergence numerically, but the Laurent series coefficients should be computed analytically.) If the Laurent series doesn t converge to Log z, what does it converge to? How does the result depend on the choice of the radius R of the contour integral in Eq. (1)? 8. (15 points) Show how to compute the integral 0 log x x 2 + 4x + 4 explicitly, by relating it to a suitable contour integral involving a suitable branch of the complex-valued function f(z) = (log z)2 z 2 + 4z + 4. Note, unlike the examples in class, f(z) is not quite the same as the complexvalued extension of the integrand in Eq. (2). Using the direct extension doesn t seem to help, as far as I can tell. But you can use similar ideas as in class to set up a contour integral of f(z) given above which will yield the value of the integral (2). 9. (10 points) Compute an analytical expression for n= ( 1) n n n 3 + 1/8. Confirm your result through numerical evaluation of your analytical expression and direct numerical computation of partial sums. 10. (10 points) Compute an analytical expression for 1 n 4 + 4n 2. Confirm your result through numerical evaluation of your analytical expression and direct numerical computation of partial sums. 11. (10 points) Show that lim N 1 N 2 4 6 2N 1 3 5 (2N 1) = π. (2) 3
12. (30 points) Consider the infinite products and ( f(z) = (1 z 2j 1 ) j=1 g(z) = (1 + z n ) ) 1 (a) (10 points) Show that each infinite product expansion converges normally (absolutely and uniformly on compact subsets) in the unit disk D = {z C : z < 1}, and therefore each defines an analytic function on D. (b) (10 points) Show that f(z) = g(z) on D. (c) (10 points) Use this fact to derive the following result from number theory: For each positive integer n, the number of ways that n can be expressed as a sum of odd positive integers is equal to the number of ways that n can be expressed as a sum of distinct positive integers. For example, the positive integer 6 can be expressed in 4 ways as the sum of odd integers: 1 + 5 = 3 + 3 = 1 + 1 + 1 + 3 = 1 + 1 + 1 + 1 + 1 + 1 and 4 ways as the sum of distinct positive integers 6 = 1 + 5 = 2 + 4 = 1 + 2 + 3. Note in particular that sums involving just one summand are to be counted. 13. (15 points) Consider the power series ( 1) n nz 2n 1. (a) (5 points) What is the largest open domain over which this power series converges? (b) (10 points) Find an analytic continuation of this power series onto all of the complex plane except for some isolated singularities. What are those singularities? 14. (25 points) Consider the function f(z) = Show that when f is analytically continued to the entire complex plane, its only singularity is a simple pole at z = 1, and its only zeros are at the nonnegative even integers and a subset of the line Re z = 1/2. n z 4
15. (20 points) Let D be the domain defined by the whole Riemann sphere except for the arc C = {z = e iθ, 0 θ π}. Find a one-to-one conformal map of this domain onto the upper hemisphere of the Riemann sphere such that the point at infinity in D is held fixed. 16. (25 points) Consider an ideal, incompressible, potential fluid flow which is bounded on one side by an infinite wall, to which is attached a cylinder which is infinitely long and has radius a. (Of course in practice, infinitely long just means very long). The cylinder is attached to the wall along one of its infinitely long straight lines (generators) on its surface. Far away from the wall, the flow is parallel to the wall and has speed V. Determine, using complex variable techniques, the detailed fluid flow pattern resulting from the cylindrical obstacle attached to the wall. 17. (20 points) We discussed in class how complex analytic functions could be used to represent ideal fluid flows which where incompressible and had no vorticity. A slight extension of the ideas permits the representation of fluid flows which have their vorticity concentrated on some thin sets. Mathematically, we consider a two-dimensional steady velocity field v(x, y) which is incompressible ( v = 0) everywhere on a simply connected fluid domain D and for which the vorticity ω = v is zero everywhere on D except on some set S. Generally, S will be a union of points and one-dimensional curves, and ω will actually be singular on S (like a Dirac delta measure). This is of course an idealization of a flow with vorticity concentrated in a small region. Because the flow is incompressible on a simply connected domain, the stream function ψ(x, y) will still be a singlevalued harmonic function. The flow potential φ(x, y), however, will usually be multivalued because v = 0 on D \S, which is usually a multiply connected domain. As a result, the complex potential Ω(z) = φ(x, y) + iψ(x, y) (with z = x + iy) will be a multi-valued function which is analytic on D except for branch points/cuts on S. The simplest example of such a flow with concentrated vorticity is the point vortex defined on the whole complex plane D = C. The point vortex is described by the complex potential where k R and z 0 C. Ω(z) = ik log(z z 0 ), (3) (a) (5 points) Compute the velocity field corresponding to this complex potential. Express it in terms of physical space coordinates, describe the general structure of the flow, and explain how the values of k and z 0 are related to the basic physical properties of the flow. (b) (10 points) The vorticity of this flow is zero everywhere except at z 0, where it is divergent. To obtain a meaningful sense of how much the fluid 5
is spinning, one therefore discusses the circulation about a simple closed curve C as the line integral Γ v ˆt ds, C where ˆt is the unit tangent vector to the curve (directed counterclockwise) and s is an arclength parameter. When the vorticity field ω(x, y) is smooth on and inside C, then the circulation can be related to the vorticity through an application of Green s theorem: Γ = ω(x, y) dx dy, where A is the A region bounded by C. But even when ω is singular (as for the point vortex), the circulation is perfectly well-defined for any simple closed curve that does not intersect the singular set S. Show that the circulation about such a simple closed curve C can be related to the complex velocity potential (in general; not just for the single vortex case (3) by Γ = Re C Ω, where the right hand side refers to the change in the value of Ω(z) when it is followed continuously once around the curve C counterclockwise. (Recall our discussion of winding number). And while you re at it, also show that the flux out of the the curve C: F v ˆn ds, C where ˆn is the unit normal vector to the curve C and s is an arclength parameter, is related to the complex potential by: F = Im C Ω. (c) (5 points) Use complex-variable techniques to calculate the circulation about an arbitrary simple closed curve not intersecting the singular vorticity set. 18. (35 points) A somewhat more interesting flow with singular vorticity is defined by the complex potential Ω(z) = ik log(z + g(z)), where k R, log is to be interpreted as a multi-valued function, and g(z) is a single-valued function defined as the branch of the function (z 2 1) 1/2 which has branch cut {z C : 1 Re z 1, Im z = 0} and assumes the value g(2) = 3. 6
(a) (5 points) Show that the singular vorticity set for this flow is exactly the branch cut of g: S = {z C : 1 Re z 1, Im z = 0}. This flow can be considered as a model for ideal flow past a plate, and as a super-idealized model for flow past an airplane wing (which does rely on the generation of vorticity about it at takeoff in order to establish and maintain lift during flight). (b) (15 points) Calculate the flow past the upper and lower part of the plate (lim y 0 v(x, y) and lim y 0 v(x, y) for 1 < x < 1). The calculation is most elegantly done by working with complex variables until the last step, so proceed this way for full credit. Be sure to explain carefully any subtle steps in your derivation! Also describe in physical terms what your results imply. (c) (5 points) Sketch a two-dimensional vector plot of the flow in physical space coordinates. (d) (10 points) Calculate the circulation of the flow about any curve C which does not intersect the singular vorticity set S (representing the plate). 19. (15 points) The function sec z = 1/(cos z) can be expanded as the following Laurent series on the indicated domains: sec z = sec z = n= n= a n z n for z < π/2, b n z n for π/2 < z < 3π/2. Obtain explicit values for the following coefficients, explaining your derivation: (a) (5 points) {a n } 2 n= 2 (b) (10 points) {b n } 2 n= 2 20. (10 points) Compute the integral 2 2 4 (4 x2 )(x + 2) 2 dx 21. (35 points) Consider the problem of determining the roots of the equation: 2z 4 + z 3 + 2z 2 + 1 = 0. (4) 7
(a) (15 points) Use purely analytical reasoning to determine how many roots lie on the coordinate axes and within each quadrant of the complex plane. You can earn bonus credit by narrowing down the location of the roots more precisely than by quadrant, using analytical arguments. (b) (20 points) Develop a computational algorithm which exploits the fact that we are seeking the roots of an entire function to compute the roots of the equation (4) more precisely. Your algorithm should not be a generic root-finding algorithm for a two-dimensional real-valued function of two real variables. 8