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Math 617
Fall 2015 Journal

December 16, 2015
The final examination was given, and solutions are available.
December 8, 2015
As indicated in the slides from class, in this final class meeting we discussed Möbius transformations (linear fractional transformations) as functions on projective space and as functions on the complex plane. In particular, we saw that Möbius transformations preserve the set of lines and circles in the plane.
As scheduled by the registrar, the final examination takes place on the afternoon of Wednesday, December 16, from 1:00 to 3:00, in the usual classroom.
December 3, 2015
As indicated in the slides from class, we covered the homotopy proof of the perturbation version of Rouché’s theorem, the notion of conformal mapping, and the definition of one-dimensional complex projective space (a new realization of the extended complex numbers).
The assignment for next time (not to hand in) is to make a list of the most important concepts and theorems from the course.
December 1, 2015
As indicated in the slides from class, we discussed Rouché’s theorem (both the perturbation version and the symmetric version), and we worked on some applications of the theorem from past qualifying examinations.
Since you conquered Rouché’s theorem in class, there is no assignment to hand in next time.
November 24, 2015
As indicated in the slides from class, we looked at Riemann’s 1859 paper about the distribution of prime numbers and discussed the Riemann zeta function and the Riemann Hypothesis.
In view of the Thanksgiving holiday, there is no assignment to hand in next time.
November 19, 2015
As indicated in the slides from class, we discussed the argument principle and looked at a concrete example of a polynomial of degree 50.
Here is the assignment to hand in next time.
November 17, 2015
As indicated in the slides from class, we discussed logarithms in the complex plane, arriving at the statement that a zero-free analytic function in a simply connected region admits an analytic logarithm.
The assignment to hand in next time is Exercise 21 from Chapter III, Section 2, page 44, which asks for a proof that there is no branch of the logarithm defined on \(\C\setminus\{0\}\).
November 12, 2015
As indicated in the slides from class, we worked through John Dixon’s proof of the homology version of Cauchy’s theorem, filling in some of the details.
The assignment to hand in next time is the following three exercises from the textbook: Exercise 7 on page 87, Exercise 4 on page 100, and Exercise 6 on page 100.
November 10, 2015
As indicated in the slides from class, we discussed the notions of winding number and simple connectivity.
The assignment for next time is to read Section 7 of Chapter IV, pages 97–99 (about counting zeroes and the open mapping theorem).
November 5, 2015
The second examination was given, and solutions are available.
November 3, 2015
As indicated in the slides from class, we reviewed for the exam to be given next class, we extended the maximum principle to the case of weak local maxima, and we proved the existence of the partial-fractions decomposition of rational functions.
October 29, 2015
As indicated in the slides from class, we proved Morera’s theorem. Also, we looked at some exercises from the textbook, including McCarthy’s proof of the Cayley–Hamilton theorem using complex analysis.
In view of the examination to be given on November 5, the assignment for next time (not to hand in) is to make a list of the most important concepts and theorems covered since the first examination.
October 27, 2015
As indicated in the slides from class, we discussed the identity principle for analytic functions and the statement of Morera’s theorem.
The assignment to hand in next time is Exercises 8 and 10 from page 80 of the textbook.
October 22, 2015
As indicated in the slides from class, we discussed Picard’s little theorem about the range of entire functions, proved Liouville’s theorem about the range of entire functions, and proved the mean-value property of analytic functions in disks (a corollary being a preliminary local version of the maximum principle).
Here is the assignment to hand in next time.
October 20, 2015
As indicated in the slides from class, we discussed the classification of isolated singularities, the Casorati–Weierstrass theorem, and Picard’s big theorem.
The assignment to hand in next time is problem 2 from the August 2014 qualifying examination (about isolated singularities and residues of two concrete functions).
October 15, 2015
As indicated in the slides from class, we extended Cauchy’s residue theorem to the case of arbitrary singularities, and we proved Riemann’s removable-singularities theorem.
Here is the assignment to hand in next time.
October 13, 2015
As indicated in the slides from class, we moved beyond starshaped regions to develop the Laurent series for functions analytic in annuli.
The assignment to hand in next time is Exercise 4 on page 110 of the textbook, which asks for Laurent expansions of the rational function \[ \frac{1}{z(z-1)(z-2)} \] in the three natural annuli centered at the origin in which the function is analytic.
October 12, 2015
I posted solutions to the first examination.
October 8, 2015
As indicated in the slides from class, we extended Cauchy’s theorem to star-shaped regions and deduced that the integral representation theorem and the residue theorem hold for circular regions.
Here is the assignment to hand in next time.
October 6, 2015
As indicated in the slides from class, we pushed Cauchy’s theory beyond rectangles. In particular, we started considering integrals over continuously differentiable curves.
The assignment to hand in next time is exercises 9 and 10 on page 67 of the textbook. The first of these exercises can be interpreted geometrically as the integral of the function \(1/z\) over the unit circle traversed \(n\) times, and the second asks for the integral of \(z^n\) over the unit circle.
October 1, 2015
The first exam was given. I will post the exam and solutions after the student who was away at a conference last week takes the exam.
September 29, 2015
As indicated in the slides from class, we reviewed for the examination to be given next class, and we applied Cauchy’s theorem for rectangles to prove that an analytic function in a rectangle has an analytic antiderivative.
September 24, 2015
As indicated in the slides from class, we discussed hypotheses guaranteeing that Cauchy’s rectangle theorem will evaluate integrals over the real axis, and we saw that infinite differentiability of analytic functions is a consequence of Cauchy’s integral representation on rectangles.
In view of the examination to be given on October 1, the assignment for next time is to make a list of the most important concepts and theorems covered so far.
September 22, 2015
As indicated in the slides from class, we examined Cauchy’s method from 1814 of integrating over rectangles, leading to a statement of Cauchy’s residue theorem for rectangles. A typical application is the evaluation of \[ \int_0^\infty \frac{\cos(x)}{x^2+1}\,dx \] by considering the related problem of integrating \[ \frac{e^{iz}}{z^2+1} \] over a rectangle and letting the three extra sides of the rectangle in the upper half-plane run off to infinity.
The assignment to hand in next time is problem 4 from the August 2012 qualifying examination, which asks for an evaluation of \[ \int_{-\infty}^\infty \frac{\cos x\,dx}{(x^2+1)(x^2+4)}. \]
September 17, 2015
As indicated in the slides from class, we discussed Wirtinger’s notation for derivatives, the relation between analytic functions and harmonic functions, and some properties of the complex exponential, sine, and cosine functions.
Here is the assignment to hand in next time.
  • Show that if \(u\colon\C\to\R\) is twice continuously differentiable (in the real sense), then \(u\) is harmonic if and only if \(\dfrac{\partial u}{\partial z}\) is analytic.
  • Exercise 6 from page 44 of the textbook, which asks for a complete solution set in the complex plane for each of the following equations:
    • \(e^z=i\)
    • \(e^z=-1\)
    • \(e^z=-i\)
    • \(\cos z=0\)
    • \(\sin z=0\)
  • Exercise 19 from page 44 of the textbook, which asks for a proof that if \(f(z)\) is analytic in a region of the complex plane, then \(\overline{f(\bar z)}\) is analytic in the conjugate-symmetric region.
September 15, 2015
As indicated in the slides from class, we discussed the notions of complex differentiability and analyticity. In particular, we derived the Cauchy–Riemann equations from the point of view of the derivative being represented by a linear transformation.
Here is the assignment to hand in next time.
  • Exercise 1 from page 43 of the textbook: namely, if \(f(z)=\left|z\right|^2 = x^2+y^2\), then \(f\) has a complex derivative only at the origin.
  • Show that the function equal to \(z^{5}\big/\left|z\right|^{4}\) when \(z\ne 0\) and equal to \(0\) when \(z=0\) is continuous, and the Cauchy–Riemann equations hold at the origin, but the function is not complex-differentiable at the origin.
September 10, 2015
As indicated in the slides from class, we discussed the radius of convergence of power series and worked on some exercises from the textbook.
The assignment to hand in next time is problem 1 from the August 2015 qualifying examination (about convergence of series) and problem 1 from the January 2015 qualifying examination (about an inequality for complex numbers).
September 8, 2015
As indicated in the slides from class, we followed up on Taylor’s observation that stereographic projections from two spheres of different radius but with the same north pole are compatible; we talked about which functions can be obtained as limits of polynomials; and we recalled some convergence tests for infinite series, all contained in Cauchy’s 1821 Cours d'analyse.
The exercise to hand in next time is to prove the following convergence test from the same book: Suppose that \(a_n \gt 0\) for every natural number \(n\), and suppose that the limit \[ \lim_{n\to\infty} \frac{\log a_n}{\log(1/n)} \] exists and equals \(h\). If \(h \gt 1\), then \(\sum_{n=1}^\infty a_n\) converges, and if \(h \lt 1\), then \(\sum_{n=1}^\infty a_n\) diverges.
September 3, 2015
In class, we discussed some similarities and differences between \(\C\) and \(\R\) and between \(\C\) and \(\R^2\), we worked in groups on an exercise about the geometry of \(\C\) (included in the slides from class), and we discussed two models of stereographic projection. (I also posted the slides from the first class.)
Here is the assignment to hand in next time.
  • Show that a real-linear transformation of \(\R^2\), represented by a real matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), corresponds to a complex-linear transformation of \(\C\) if and only if \(a=d\) and \(b=-c\).
  • Derive the following formulas for the model of stereographic projection that uses a sphere of diameter \(1\) tangent to the complex plane at the origin (a different model from the one in the textbook).
    If \(x+iy\) (or \(z\)) is a point in the complex plane, and \( (x_1,x_2,x_3)\) is the corresponding point on the sphere, then \[ x=\frac{x_1}{1-x_3} \qquad \text{and} \qquad y=\frac{x_2}{1-x_3}; \] also \[ x_1=\frac{x}{\left|z\right|^2 +1} \qquad \text{and} \qquad x_2 = \frac{y}{\left|z\right|^2 +1} \qquad \text{and} \qquad x_3 = \frac{\left|z\right|^2}{\left|z\right|^2 +1}. \]
    Moreover, the spherical distance \(d\) for this model has the following properties: \[d(z,z') = \dfrac{\left|z-z'\right|} {\sqrt{\left|z\right|^2 +1}\,\sqrt{\left|z'\right|^2+1}} \quad\text{and}\quad d(z,\infty) = \frac{1}{\sqrt{\left|z\right|^2 +1}}.\]
September 1, 2015
In the first class meeting, we did introductions and discussed various characterizations of the complex numbers.
The standing assignment is to read the textbook. Additionally, solve the following two exercises and hand in your solutions next class (Thursday).
  • Represent complex numbers \(z\) and \(w\) by vectors \((x,y) \) and \((u,v)\) in \(\R^2\). How is the complex number \(\overline{z} w\) related to the usual dot product and cross product of the vectors \((x,y) \) and \((u,v)\)? [The cross product lives in three dimensions, but you can view the vectors as living in \(\R^3\) by treating them as \( (x,y,0)\) and \( (u,v,0)\).]
  • Exercise 1 on page 4 of the textbook, which asks for a proof of the inequality \[ \bigl| \left|z\right|-\left|w\right|\bigr| \le \left|z-w\right| \] (a version of the triangle inequality), along with necessary and sufficient conditions on the complex numbers \(z\) and \(w\) for equality to hold.
August 31, 2015
The final examination schedule became available at the registrar’s website, and I added the date and time of the final exam (1:00–3:00 in the afternoon of Wednesday, December 16) to the course home page.
August 27, 2015
This site went live today.
I look forward to seeing you at the first class meeting on Tuesday, September 1. Once the semester starts, watch this page for regular updates about assignments, what we did in class, and so forth.