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Math 617
Fall 2014 Journal

December 12
The final examination was given.
December 9
Continuing the discussion of Möbius transformations, we investigated the cross ratio invariant.
December 4
In class, we discussed one-dimensional complex projective space as a model of the extended complex numbers. We saw that the space supports the structure of a complex manifold, and the linear structure on \(\C\times\C\) induces the group of linear fractional transformations (Möbius transformations).
There is no assignment to hand in next time. A preview of the final exam is available.
December 2
In class, we discussed stereographic projection and a connection with geometric inversion in three dimensions. We worked in groups on computing formulas for stereographic projection in the two standard models: one with the plane cutting the sphere at the equator, and one with the sphere tangent to the plane.
Here is the assignment to hand in next time.
  1. The spherical distance \(d(z,w)\) between complex numbers \(z\) and \(w\) is the Euclidean distance between the image points on the sphere under stereographic projection. A computation shows that \[ d(z,w) = \frac{\left|z-w\right|} {\sqrt{1+|z|^2} \sqrt{1+|w|^2}} \] for the tangent model of stereographic projection, and \(d(z,w)\) equals twice that expression for the equatorial model of stereographic projection.
    Question for you to answer: How should \(d(z,\infty)\) be defined?
    We previously said that \(z\to\infty\) in the complex numbers if and only if \(\left|z\right|\to\infty\) in the real numbers. Now we can say that \(z\to\infty\) if and only if \(d(z,\infty)\to 0\).
  2. In a previous exercise, you showed that the holomorphic automorphism group of the entire plane is generated by rotations, dilations, and translations. Your new task is to show that the holomorphic automorphism group of \(\C\setminus\{0\}\), the punctured plane, is generated by rotations, dilations, and inversion. (I stated this proposition in class last time, and the problem is also Question 3 on the August 2011 qualifying exam.)
    Hint: An automorphism of the punctured plane has an isolated singularity at the origin. Show that if the singularity is removable, then the automorphism extends to be an automorphism of the entire plane fixing the origin. Show that if the singularity is a pole, then composing with inversion (that is, taking the reciprocal) reduces to the preceding case. Finally, show that an automorphism of the punctured plane cannot have an essential singularity.
November 25
In class, we discussed conformality of holomorphic functions having nonzero derivative, and we looked at geometric properties of inversion.
There is no assignment to hand in next time. Enjoy the Thanksgiving holiday!
November 20
In class, we discussed the local behavior of holomorphic functions: namely, holomorphic functions are essentially power functions locally.
Here is the assignment to hand in next time.
  1. Suppose \(f\) is holomorphic on the unit disk, and \( \left|f'(z)-f'(0) \right| \lt \left|f'(0) \right|\) when \(0\lt \left|z\right| \lt 1\). Prove that \(f\) is an injective function on the unit disk.
    Hint: Either use that \[ f(z_2)-f(z_1) = \int_{z_1}^{z_2} \{ f'(z) - f'(0)\} + f'(0)\,dz, \] or apply Proposition 7.3 on page 107.
  2. Prove that every injective entire function has the form \(z\mapsto az+b\) for some complex constants \(a\) and \(b\), where \(a\ne 0\).
    Suggestion: If the entire function is a polynomial, apply the fundamental theorem of algebra. On the other hand, if \(f(z)\) is a nonpolynomial entire function, then \(f(1/z)\) has an essential singularity at the origin. Invoke the Casorati–Weierstrass theorem (which is the standard name for Proposition 3.14(ii) on page 45).
    (The conclusion follows more easily from the big Picard theorem, but invoking that deep theorem from Chapter 20 is not quite fair.)
  3. Suppose that \[f(z)= \frac{z}{(1-z)^2}\] when \(\left|z \right|\lt 1\). Prove that \(f\) is injective on the unit disk.
    Hint: You can solve this problem without a hint.
    (But if you are really stuck, then google “Koebe function”.)
November 18
In class, we discussed the open mapping theorem for holomorphic functions. Then we worked in groups on exercises from the end of Chapter 5.
The assignment to hand in next time is your group's choice of two of the exercises. You should submit one paper per group.
November 13
I posted solutions to the second exam. There was a mistake in the initial posting, now corrected.
In class, we discussed Rouché's theorem and looked at examples.
The assignment to hand in next time is your choice of two of the following three problems.
November 11
In class, we discussed the argument principle and as an example showed that the polynomial \(z^{50}+z+1\) has exactly \(12\) zeroes in the first quadrant.
Here is the assignment to hand in next time: one problem with several steps.
  • Apply the argument principle to show that the polynomial \(z^3-3z^2+4z-1\) has no zero in the second quadrant. Deduce that there is also no zero in the third quadrant.
  • Apply the intermediate-value theorem (from real calculus) to show that the polynomial has a zero on the positive part of the real axis.
  • Deduce further that the polynomial has one zero in the open first quadrant and one zero in the open fourth quadrant.
Remark: This exercise is a toy problem to practice technique. The zeroes of a cubic polynomial are, of course, available through an exact formula.
November 6
The second examination was given.
There is no assignment to hand in next time.
November 4
In class, we discussed residues, l'Hôpital's rule for holomorphic functions, a counterexample to the mean-value theorem in the setting of holomorphic functions, and Picard's great theorem about essential singularities.
In view of the exam taking place on Thursday, there is no assignment to turn in on that day or on the following Tuesday.
October 30
In class, we worked in groups on the evaluation of real definite integrals via residues.
The assignment to hand in next time is your choice of two of the following problems about residues and integrals.
October 28
In class, we discussed residues and the residue theorem. The famous article of H. Pétard titled A Contribution to the Mathematical Theory of Big Game Hunting appeared as a sidelight; Method 8 in that paper is an application of the Cauchy integral formula.
Here is the assignment to hand in next time.
  1. Determine the residue of the function \(\cot(z)\) [that is, \(\cos(z)/\sin(z)\)] at each singular point.
  2. In real calculus, you learned the method of decomposing a rational function into “partial fractions,” probably without proof. Your task is to supply a proof, as follows. Suppose that \(R(z)\) is a rational function whose denominator has larger degree than the numerator. Suppose the singularities of \(R(z)\) are \(z_1\), \(\ldots\), \(z_n\), and let \(P_j\) denote the principal part of the Laurent series of \(R\) in a punctured disk centered at \(z_j\). Show that \(R(z)\) is identically equal to \(\sum_{j=1}^n P_j(z)\).
    Hint: The difference has only removable singularities and tends to zero when \(\left|z\right|\to\infty\).
October 23
In class, we discussed winding numbers, the homology version of Cauchy's theorem and Cauchy's integral formula, and the existence of Laurent series. John D. Dixon's 1971 paper “A brief proof of Cauchy’s integral theorem” is well worth reading.
Here is the assignment to hand in next time.
  1. Exercise 4.19 on page 76 in the textbook.
  2. Exercise 4.20 on page 76 in the textbook.
  3. Problem 2 on the August 2012 qualifying exam, which asks for the Laurent series in powers of \(z-1\) and \(1/(z-1)\) for the function \(z^{-1}(z+1)^{-1}\) in the annulus where \(1\lt \left|z-1 \right|\lt 2\).
October 21
In class, we discussed homotopy, simple connectivity, and the homotopy version of Cauchy's theorem.
Here is the assignment to hand in next time.
  1. Find an example of an unbounded simply connected subset of \(\C\) whose complement has two components.
  2. Exercise 4.22 on page 77 in the textbook.
October 16
In class, we discussed logarithms in the setting of complex numbers and functions.
Here is the assignment to hand in next time.
  1. Show that there is a branch of the logarithm defined on \(\{\,z\in\C: \Re(z)\gt 0\,\}\), the right-hand half-plane, that maps this half-plane bijectively to the horizontal strip \(\{\,z\in\C: \left|\Im(z)\right|\lt \pi/2\,\}\).
  2. Exhibit an entire function \(f\) such that \(\left|f(z)\right|\le 1\) when \(\left|\Im(z)\right|=1\), yet \(\left|f(z)\right|\) is unbounded on the real axis (where \(\Im(z)=0\)).
    [This example shows that there is no global version of the maximum principle unless some extra hypothesis is included concerning the behavior of the function “at infinity”.]
  3. Evaluate \(i^{(i^i)}\) and \( (i^i)^i\) using the principal branch of the logarithm. Then find all possible values of these expressions for an arbitrary branch of the logarithm.
October 14
In class, we worked in groups on various exercises from Chapter 3, and we discussed the maximum principle (local and global versions) and the Schwarz lemma.
The assignment to hand in next time is to append solutions to two additional exercises to the three that you have already done.
October 9
In class, we discussed convergence and uniform convergence of power series, isolated zeroes of holomorphic functions, and singularities.
The assignment is to select three problems from Exercises 3.6–3.20 on pages 47–50 and to solve them.
October 7
We followed up on the exam, discussing such topics as the implicit-function theorem, the prime-number theorem, small gaps between prime numbers, and Morera's proof of his theorem. Also we discussed Cauchy's estimates for derivatives and the preservation of holomorphy under uniform convergence.
The assignment to hand in next time is Exercises 3.4 and 3.5 on page 47. These exercises have to do with Liouville's theorem and a generalization. (The conclusions of both exercises can be strengthened.)
October 2
The first examination was given, and solutions are available.
September 30
By way of preparing for the exam to be given on Thursday, we looked at an old exam. And we applied Cauchy's integral formula to prove Liouville's theorem (a different proof from the one in the textbook).
In view of the exam taking place on Thursday, there is no assignment to turn in on that day or on the following Tuesday.
September 25
In class, we discussed a basic version of Cauchy's integral formula.
Some mathematicians play a game of trying to apply each theorem of complex analysis to prove the fundamental theorem of algebra: namely, that every nonconstant polynomial has a root in the complex numbers. Your assignment to hand in next time is to expand the following sketches to provide three different proofs of the fundamental theorem of algebra. All three proofs use the method of reductio ad absurdum, the counterfactual hypothesis being that \(p(z)\) is a polynomial of degree \(n\) (where \(n\ge 1\)) that is nonzero for every complex number \(z\).
  1. Provide justification for each step in the following argument: \[ 0 = \oint_{|z|=R} \left( \frac{p'(z)}{p(z)} -\frac{n}{z}\right) + \frac{n}{z} \,dz = 2\pi i n + O(1/R), \] so a contradiction arises when \(R\to \infty\).
  2. First solve Exercise 2.6 on page 28 of the textbook, which says that the value of a holomorphic function at the center of a disk equals the average of the values of the function around the boundary circle. Then apply the exercise to argue that \[ \frac{1}{p(0)} = \frac{1}{2\pi} \int_0^{2\pi} \frac{1}{p(re^{it})} \,dt = O(1/r^n), \] so a contradiction arises when \(r\to \infty\).
  3. Show that there is a polynomial \(Q(z,w)\) of two complex variables \(z\) and \(w\) such that \(Q(z,w) = w^n p(z/w)\) when \(w\ne 0\). (The polynomial \(Q\) is uniquely determined by \(p\).) Moreover, show [under the hypothesis that \(p(z)\) is never equal to \(0\)] that \(Q(z,w)\) is equal to \(0\) if and only if \( (z,w)\) is the point \((0,0)\) in \(\C^2\). Argue that the integral \[ \oint_{|z|=1} \frac{p'(z)} {Q(z,w)} \,dz \] is a continuous function of \(w\) that equals \(0\) when \(w\ne 0\) but equals \(2\pi i n\) when \(w=0\); contradiction.
    [Part of the interest of this third proof is that no growth estimate for polynomials is required.]
September 23
In class, we discussed Wirtinger's notation for complex partial derivatives and some ideas in the proof of Morera's theorem and in Goursat's proof of Cauchy's theorem.
The assignment to hand in next time is as follows.
  1. Solve Exercise 2.5 on page 26 of the textbook.
  2. Evaluate \(\int_C (1/\,\overline{z}\,)\,dz\), where \(C\) is the unit circle oriented in the usual counterclockwise direction, and \(\overline{z}\) is the complex conjugate of the complex variable \(z\).
September 18
The discussion in class followed up on Cauchy's integral theorem, touching on such items as the path-deformation principle, Morera's converse to Cauchy's theorem, and Goursat's refinement of Cauchy's theorem.
The assignment to hand in next time is as follows.
  1. Solve Exercise 2.1 on page 16 of the textbook.
  2. In his 1851 dissertation, Grundlagen für eine allgemeine Theorie der Functionen einer Veränderlichen complexen Grösse, Riemann states in section 7 the following proposition, which nowadays we call “Green's theorem in the plane”: \[ \iint \left( \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} \right) \,dT = - \int (X \cos \xi + Y \cos\eta)\,ds. \] Here the capital letters \(X\) and \(Y\) denote functions, the lowercase letters \(x\) and \(y\) are the standard real coordinates, the symbol \(dT\) represents the two-dimensional area element in some region of the plane, the symbol \(ds\) represents the arclength element on the boundary curve, and the Greek letters \(\xi\) (xi) and \(\eta\) (eta) denote the angles made with the \(x\) and \(y\) axes by the inner-pointing normal vector to the boundary curve. Verify that Riemann's statement is equivalent to our usual version of Green's theorem: in particular, check that the plus and minus signs are right.
  3. Prove that \[ \left| \pi - \int_{-r}^r \frac{\sin x}{x} \,dx \right| \lt \frac{4}{r} \] for every positive number \(r\), and deduce that \[ \int_{-\infty}^\infty \frac{\sin x}{x}\,dx = \pi. \] (The number \(4\) in the inequality is not the best possible: you might find a smaller constant that works.) Suggestion: Observe that \(\dfrac{\sin x}{x}\) is the imaginary part of \(\dfrac{e^{ix}-1}{x}\) when \(x\) is real. The function \(\dfrac{e^{iz}-1}{z}\) can be viewed as a function that is holomorphic in the whole plane (namely, looking at the Maclaurin series of the numerator reveals that there is a “removable singularity” when \(z=0\)). Hence Cauchy's integral theorem applies. Integrate this function of \(z\) on a simple closed curve that runs along the real axis from \(-r\) to \(r\) and then joins \(r\) to \(-r\) in the upper half-plane (for instance, a triangle or a rectangle or a semi-circle). You can find the exact value of the integral of \(1/z\) over a curve in the upper half-plane starting at \(r\) and ending at \(-r\). What remains is to estimate the integral of \(e^{iz}/z\) on such a curve by applying not Lemma 2.1 but instead the inequality on the top of page 19.
September 16
In class, we followed up on convergent power series, which locally are the same as holomorphic functions, and we saw that Green's theorem in the plane implies a version of Cauchy's integral theorem.
The assignment to hand in next time is as follows.
  1. Solve Problem 1 on the January 2013 qualifying exam, which asks for a proof that the coefficient sequence of the Maclaurin series of the rational function \(z/(1-z-z^2)\) is the sequence of Fibonacci numbers.
  2. The Note at the top of page 19 in the textbook proves that \[ \left| \int_a^b f(t)\,dt \right| \le \int_a^b \left|f(t)\right|\,dt \] when \(a\lt b\) and \(f\colon [a,b]\to\mathbb{C}\) is a continuous function. When does equality hold in this inequality?
September 11
In class, I augmented the discussion of the Cauchy–Riemann equations by mentioning the Looman–Menshov [Menchoff] theorem. Then we discussed some convergence tests for infinite series and Cauchy's formula for the radius of convergence of a power series.
The assignment to hand in next time is as follows.
  1. Suppose \[ f(z) = \begin{cases} \exp(-1/z^4),& z\ne 0,\\ 0, & z=0. \end{cases} \] Show that \(f\) satisfies the Cauchy–Riemann equations at every point of the plane, yet \(f\) fails to be holomorphic in a neighborhood of the origin. (This example is due to H. Looman.)
  2. Show that the root test is a more discerning test than the ratio test. Namely, show that if \((a_n)\) is a sequence of positive real numbers such that \(\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}\) exists, then \(\lim\limits_{n\to\infty} a_n^{1/n}\) exists too and has the same value. Find an example of a sequence \((a_n)\) of positive real numbers for which \(\lim\limits_{n\to\infty} a_n^{1/n}\) exists but \(\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}\) does not exist.
  3. Determine all values of the complex number \(z\) for which the power series \( \sum_{n=1}^{\infty} 3^{n} z^{3^{n}}\) converges. (Notice that this series is a gap series: many of the powers of \(z\) have coefficient equal to \(0\).)
  4. Dirichlet's convergence test says that if \( (a_n)\) is a monotonically decreasing sequence of real numbers with limit equal to \(0\), and if \( (b_n)\) is a sequence of (either real or complex) numbers with bounded partial sums [that is, there is a constant \(M\) such that \(\left|\sum_{n=1}^N b_n\right| \le M\) for every natural number \(N\)], then the infinite series \(\sum_{n=1}^\infty a_n b_n\) converges. [You can read the proof, based on Abel's method of partial summation, at various sites online, for instance at wikipedia. Dirichlet's test generalizes the more familiar alternating-series test.]
    Here is your task: Apply Dirichlet's test to verify the claim [stated in class and also on page 10 of the textbook] that the series \(\sum_{n=1}^\infty z^n/n\) converges at every boundary point of the unit disk except the point where \(z=1\).
September 9
We discussed continuity, differentiability, and the Cauchy–Riemann equations.
The assignment to hand in next time is Exercises 0.3 and 0.7 on pages 6–7 in the textbook.
September 4
We discussed functions and limits in the setting of the complex plane.
The assignment to hand in next time is as follows.
  1. Prove that \(\lim_{n\to \infty} e^{in}\) does not exist. (The variable \(n\) is understood to be a natural number.)
  2. Prove that \(\lim_{y\to\infty} \frac{1}{\sin(iy)} =0\). (The variable \(y\) is understood to be real.)
  3. Solve problem 6 on the August 2014 complex analysis qualifying examination, which asks for a proof that \[ \tfrac{1}{4} |z| \lt \left|1-e^z\right| \lt \tfrac{7}{4} |z| \] when \(0\lt |z| \lt 1\).
September 2
At the first class meeting, we discussed definitions and properties of the complex numbers: an algebraically closed field that lacks a natural order relation. We looked at some examples of disks, half-planes, and conic sections defined by relations involving the absolute values and the real parts of complex expressions.
The assignment to hand in next time is the following exercise. Complex numbers \(z\) and \(w\) can be represented either as expressions \(x+iy\) and \(u+iv\) or as vectors \( (x,y)\) and \( (u,v)\). What is the relation between the product \( \overline{z}w\) (where \(\overline{z}\) is \(x-iy\), the complex conjugate of \(z\)) and the scalar product (dot product) and vector product (cross product) of the vectors representing \(z\) and \(w\)?
August 6
I look forward to seeing you at the first class meeting on September 2. Once the semester starts, watch this page for regular updates about assignments, what we did in class, and so forth.