Standard Generals Questions - Complex Analysis Compiled by V. Rytchkov Conformal mappings ------------------ What is the Riemann mapping theorem? Why isn't the complex plane conformally equivalent to the unit disk? Why are all automorphisms of the unit disk linear fractional transformations? What can you say about conformal mapping of multiply connected regions? How many degrees of freedom in choosing non-conformally-equivalent annuli with slits missing. What is special about a conformal map? When are two annuli conformally equivalent? Can you map an open square to an open disc analytically? What about continuous extension to the boundary? Talk about isometries of Lobachevsky plane. Talk about conformal isomorphisms of compact Riemann surfaces. Which of them have an infinite group. Draw one doubly connected region in the plane. Now another. Can they be mapped conformally to one another? Prove that such a region can be mapped conformally to an annulus (if no component of the complement in the sphere is a single point...). Now what about triply connected regions? Can they be canonically mapped to a circle with circular holes punched? What does "simply connected" mean? State and prove the Riemann Mapping Theorem. You used the existence of square roots of holomorphic functions with no zeros on a simply connected domain. Why can you do this? What happens at the boundary in the Riemann Mapping Thm? Say in the case of a polygon mapping to the half-plane. State and prove the Schwarz reflection principle. What does it look like at the corners? Say it's an equilateral triangle. Give a conformal map of the right half plane onto the unit disc. What can you say about an entire function that maps into the plane minus the negative real axis? What about the plane minus a segment? Can you generalize the above cases? (i.e. Prove the small Picard Theorem.) Basic facts ----------- Prove Cauchy's Thm. What theorem of multivariable calculus is this similar to? What is Stokes Thm? Prove the Cauchy Integral Formula. Prove Liouville's Thm. How many zeros can an analytic function have in the disk? Can the zeros have a limit point on the boundary? Give an example. Prove that an analytic function has an expression as a power series. State Morera. Suppose that you have a sequence of analytic functions on a region and they converge pointwise on some simple closed curve inside the region. What can you say about the sequence of derivatives on the interior of the curve? Vitali convergence theorem. Montel theorem. Uniqueness theorem (all with proof) What is a winding number? What sort of number is it? How do you prove it's an integer? How do you know that the inverse of a 1-1 holo fn is holo? Show that a holomorphic function always has a taylor series around every point. How would you prove that a C^1 function which is complex differentiable has actually infinitely many derivatives? Under what conditions is the point-wise limit of analytic functions analytic? Give an example of a real-valued real analytic function on R whose power series at zero does not represent it everywhere. State Arzela-Ascoli thm. If you know functions (holomorphic) values on boundary of region how would you give estimates on nth derivative? Why can't 1/z on unit circle extend to holomorphic function on disk? Given a complex function f on the boundary of the unit circle can you tell when it can be analytically extended inside. If f is real on the boundary when it can be represented as |g(z)|^2 where g is analytic in the unit disk? Entire and meromorphic functions -------------------------------- Define "meromorphic function". Give an example of a nonrational meromorphic function. What are the poles and residues of 1/sin(z)? How many zeros can an entire function have? Can the zeros be ANY set with no limit point? How is this related to the order and genus? Suppose we have an entire function with a poles at 1 and 2i. Given a power series for this at zero, where does it converge? How many power series are there? Why must there be more than one? How can we compute the coefficients for each? What can you say about the zeros of an entire function? Why must the sequence of zeros go to infinity? Suppose I give you a sequence of points converging to infinity. How would you construct an entire function with those points as zeros? How about doing it for the Gamma function? Talk about field of meromorphic functions on compact surfaces. Are Mer(S^2) isomorphic to Mer(T^2)? Which meromorphic functions (on some open plane set) have meromorphic antiderivatives? What is the Gamma function (give the infinite product definition) What is the integral definition? Prove the Functional equation. Riemann Zeta function. Do you know the functional equation? Talk about the meromorphic extension to the whole plane. How many zeros must an entire function of order 1/2 have? Calculus of residues -------------------- Compute the Fourier transform of 1/(1+x^2). How would you integrate $\int_{-\infty}^\infty\frac{dx}{(x^4+1)^2}$? (concepts, not calculations) Tell about integrating 1/sqrt((1-x^2)(1+x^2)) (concepts, not calculations). Can you write down a Cauchy formula for the inverse? What is Brouwer's fixed point theorem in D^2 case? What if, instead of that, I give you that |f| < 1 in the unit disc? (use Rouche) Integrate 1/(1 + x^4) from -infinity to infinity. Residue Theorem? How is it related to index of a vector field? What is the index of a vector field? How do you prove the residue theorem? Given the values of a meromorphic function on a curve, can you tell me the multiplicities of the zeroes and poles inside? Integrate (Sin[x]/x)^2. If f is meromorphic, what is the meaning of the contour integral of f'/f? There's a famous function with zeros at the negative integers, how can you express this function as a product? Singularities ------------- Classify the types of singularities. Describe each. How does a function behave near an essential singularity? (it goes crazy) In what sense? (big picard) What happens in the neighborhood of a pole? (f -> oo) In which directions does it go to infinity? Small and Big Picard. Radius of convergence of taylor series of 1/x^2+1 at 100. Consider a function analytic in the unit disc satisfying f(2z) = f(z)/(1+f(z)^2) and f(0)=0. If such a function exists can it be continued to a meromorphic function on C? Does such a function exist? If an analytic function is represented by a power series with radius of convergence one, can it ever be continued to a neighborhood of the disc in which the series converges? An. continuation, Many-valued an. functions and Riemann surfaces ---------------------------------------------------------------- What happens near z=0 for the function (log z) / z? Is z=0 a singularity? What is it? Talk about the singularity log has at 0. Discuss Riemann surfaces... Things like: "Define Riemann surface." "Define branch point." "Define sheet." "What happens when you wind around a branch point?") Talk about the analytic continuation of your favorite transcendental function. (I wrote down 'log' but was interrupted.) No, that's too simple. What about the zeta function? Define gamma function. prove functional equation. is analytic continuation unique? What is the Riemann surface of sqrt(z), over the disc and over the Riemann sphere? Give an example of a map from the torus to the Riemann sphere branched over 4 points. Say you have a Riemannian surface (i.e., Riemannian manifold of dimension 2). How can you connect it to complex analysis? (conformal coordinates give it a complex structure, so it becomes a Riemann surface.) State the theorem of uniformization for Riemann Surfaces. What would uniformization say in connection with the Riemannian surface above? More specifically, say you have a Riemannian surface, diffeomorphic to the sphere, with a metric g. What does uniformization let you conclude? Take f entire. What are the conditions for the existence of a square root function? Look at the case where the region is bounded and simply connected. What are the conditions there? How would you extend a locally constructed square root to all of the region? Talk about analytic continuation on simply connected regions. Next, you can do the same problem by using Weierstrass factorization.... Can you take the square root of f(z)=z(z-1)(z-2). Can you draw a picture to illustrate what "genus" means? (I drew some tori with different numbers of holes.) What is the Riemann surface associated to an elliptic curve? How would Riemann have answered this question? What's the fundamental group of a torus? (...homomorphism from the fundamental group to C given by integrating a differential, giving an isomorphism with C mod a lattice.) How would Weierstrass describe this isomorphism? Draw a Riemann surface in the polygonal representation. Let f be a meromorphic function on it. How can I define a log f? Now let g be another meromorphic function on here. What's \int log f dlog g? Compute it the other way, directly. What is the universal cover of C minus a point? minus two points? Classify all Riemann surfaces with first betti number equal to one. State and prove Montel's theorem. Prove it for the maps from D to C-{0,1}. Give a hyperbolic metric on say C-{0,1}. Prove any Riemann surface diffeo to sphere is biholo. with Riemann sphere. Prove uniformization (I don't know. It's very difficult!) Can every surface be conformally mapped to the complex plane(locally)? Maximum-modulus principle ------------------------- Hadamard 3-circles theorem, generalize to annuli with slits missing. An entire function f has Re(f) + Im(f) bounded. What can you say about f? If f : C -> C is analytic and bounded, what happens? What if f is entire but only Re f is bounded? Suppose you have a holomorphic function in a strip, continuous and bounded in absolute value by 1 on the boundary, and bounded everywhere. What can you say? What might one use Phragmen-Lindelof for? Elliptic functions ------------------ Talk about doubly periodic functions on C. Prove that the sum of the residues of such a function in a period parallelogram is 0. Do you know about elliptic functions? Weierstrass rho-function, differential equation for it. Miscellaneous ------------- Tell us about the Dirchlet Principle. Does it relate to a variational problem? What is a Green function? Can you have a harmonic function on the sphere with just a logarithmic singularity at infinity? What do you know about several complex variables (polydiscs, Cauchy formula, domain of holomorphy, extension from a polydisc with a hole into the whole polydisc) Do you know anything about degrees of maps or integral representations in several cplx variables? Given several quarter-circles with various holes chopped out, and values assigned to different pieces of the boundary, do there exist harmonic functions on these domains with these boundary values? Can you describe an open region of C^2 such that every holomorphic function defined there extends to a larger region? If you have real function on bdry of domain how would you "extend" it to be real part of a holomorphic function on domain? (Do for U say, discuss Poisson kernel and harmonic conjugates) If f holomorphic on C\{0} when is it a derivative of F holomorphic in C\{0} (Give answer in terms of Laurent series) Fn :D---C a seq. of holo. with L1 norm bounded by one. Show that it has a subsequence converging uniformly on compacts on disk of radius 1/2. Suppose an entire function is real on the unit circle, what does this say about the function? Suppose f:unit disc --> C has the value 5 on the line x=y. If f is holomorphic, what is f? How do you prove this? What if f is harmonic? Give a nonconstant example. Does there exist holomorphic f:unit disc --> C which has zeros at {n/(n+1): n a positive integer}? Can such an f be bounded? Do you know Blaschke condition? Let U be a region. Let H denote the set of all analytic functions on U whose L2 norms are finite. Is H a Banach space? Why? Does there exist a harmonic function in the upper half plane such that boundary value : [0,1] = 1, others 0? If so, give an example.