In the final class meeting of the semester, we discussed several topics connected with the zero-counting integral
\[\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}\,dz:
\]
namely, multiple zeroes and the discriminant of a polynomial, the open mapping theorem, and the maximum principle.
Reminders:
The final exam takes place on Friday, December 7, at 3:00 in the afternoon.
Office hours this week:
2:00–3:00pm, Wednesday, December 5
2:00–3:00pm, Thursday, December 6
10:00am–noon, Friday, December 7
The end-of-term course evaluations are available to be completed online.
Thursday, November 29
We discussed Rouché's theorem, and we worked in groups on the following three problems from past qualifying examinations.
August 2008 problem 6: If \(\alpha>1\), then the equation \[\sin z=e^\alpha z^3\] has exactly three solutions in the unit disk.
August 2009 problem 4: If \(a>0\) and \(b>2\), then the equation \[ az^3-z+b=e^{-z}(z+2)\] has exactly two solutions in the right-hand half-plane.
August 2012 problem 7: If \(\lambda>1\), then the equation \[e^z-z=\lambda\] has exactly one solution in the left-hand half-plane.
The assignment for next time (not to hand in, since next class is the final one for the semester) is to finish solving these three problems.
Tuesday, November 27
We discussed the argument principle and used it to show that the polynomial \(z^{50}+z+1\) has exactly \(12\) zeroes in the first quadrant.
The assignment to hand in next time is to show that
there is exactly one zero of the polynomial \(z^{50}+z+1\) in the subset of the first quadrant where \(0\le \arg(z)\le 2\pi/50\), and
every zero of the polynomial \(z^{50}+z+1\) in the right-hand half-plane has modulus (slightly) larger than \(1\).
Tuesday, November 20
We discussed the previous assignment, and we made a preliminary analysis of the location of the zeroes of the polynomial \(z^{50}+z+1\) as motivation for the argument principle.
There is no assignment to hand in next class. Enjoy the Thanksgiving holiday!
Thursday, November 15
In class, we discussed the general problem of integrating a rational function \(r(x)\), and we proved that when \(\int_0^\infty r(x)\,dx\) converges, the value equals the negative of the sum of all the residues of \(r(z)\log(z)\), where the logarithm is determined by a branch cut along the positive part of the real axis.
Here is the assignment to hand in next time.
Prove that when \(0\lt s \lt 1\), the improper integral
\[
\int_0^\infty x^{s-1} \cos(x)\,dx
\]
converges and equals \(\Gamma(s)\cos(\pi s/2)\), where \(\Gamma\) is the gamma function that we studied previously. Notice that convergence of the integral is an issue both at \(0\) and at \(\infty\).
Suggestion:
Integrate \(z^{s-1}e^{iz}\) around a contour that is a triangle with
vertices at \(0\), \(R\), and \(iR\), except for a circular
cutout around the origin. Then pass to the limit.
Solve problem 6 on the August
2011 qualifying exam: namely, show that
\[
\int_0^\infty \frac{x^2}{1+x^5}\,dx = \frac{\pi/5} {\sin(2\pi/5)}.
\]
Suggestion: In principle, the problem can be solved by applying a
theorem from today's class stating that this integral equals the
negative of the sum of the residues of the function \(z^2 \log(z)/(1+z^5)\). But I do not recommend that method for this particular problem, because you would have to add up five residues. A simpler method—involving only one residue calculation—is to
integrate the function \(z^2/(1+z^5)\) around a piece of pie with angle
\(2\pi/5\).
Supplementary Exercise 1 on page 75, which asks for a
computation of the
principal value of the divergent improper integral
\[
\int_{-\infty}^\infty \frac{1}{x^3-x^2+x-1}\,dx.
\]
(In class, we did not discuss Cauchy's notion of the principal value of a singular integral, but page 74 in the textbook explains the notion.)
Remark: The integral converges at \(\pm\infty\) because of the
relatively fast decay of the integrand. The principal value gets
involved because the integrand has a singularity on the integration
path at the point where \(x=1\).
Answer: The Cauchy principal value of this integral (which is known to WolframAlpha)
equals \(-\pi/2\).
Tuesday, November 13
We discussed some of the problems from the current assignment, and we worked in groups on showing that
\[
\int_0^\infty \frac{\sqrt{x}} {1+x^2}\,{dx} = \frac{\pi}{\sqrt{2}}
\]
by using a semicircular contour with a semicircular cutout around the origin.
The assignment to hand in next time is to complete this computation.
Thursday, November 8
We discussed logarithms of functions and the definition of \(z^w\) as \(e^{w\log(z)}\). We also learned from the internet the etymology of Napier's neologism “logarithm” (1614). The Oxford English Dictionary comments, “Napier does not explain his view of the literal meaning of logarithmus. It is commonly taken to mean ‘ratio-number’, and as thus interpreted it is not inappropriate, though its fitness is not obvious without explanation.”
Here is the assignment to hand in next time.
WolframAlpha evaluates the expression \(i^i\) as \(e^{-\pi/2}\). What are all the other possible values of \(i^i\)?
The point of this question is to understand how complex conjugation interacts with logarithms. Is it correct to say that
\[
\overline{\log(z)} = \log\left(\overline{z}\right)?
\]
In calculus, you learned that the expression \(x^x\) approaches \(1\) when \(x\) tends to \(0\) through positive real values. Can you say analogously that when \(z\) is a complex variable, the expression \(z^z\) approaches \(1\) when \(z\) tends to \(0\) (as a two-dimensional limit)?
Find the Taylor series centered at \(-1\) for some branch of \(\log(z)\) that is holomorphic in a neighborhood of \(-1\). What is the radius of convergence of this series?
Explain how to construct a nonvoid open set \(D\) satisfying all of the following properties.
\(D\) is a subset of the annulus \(\{\, z\in\mathbb{C}: 1\lt |z| \lt 2\,\}\).
\(D\) is a connected set, and so is the boundary of \(D\).
A holomorphic branch of \(\log(z)\) can be defined on \(D\).
The quantity \(\left|\log(z)\right|\) (the modulus of the logarithm) is unbounded on \(D\).
Tuesday, November 6
In class, we discussed extending the natural logarithm function to the complex plane.
Here is the assignment to hand in next time.
Using the principal value of the logarithm (\(-\pi\lt \arg(z)\lt \pi\)), evaluate \( \log\{i(i-1)\}\) and compare with \(\log(i) + \log(i-1)\).
Find all possible values of the expression \(\log(e^i)\) (which is Supplementary Exercise 1c on page 83 of the textbook). How does this set of values compare with the set of possible values of \(i\log(e)\)?
Thursday, November 1
We discussed the connection between residues and the method of partial fractions, the computation of residues at simple poles, and the issues in solving Exercise 9.1 in the textbook. Here is the assignment to hand in next time.
Use residues to show that
\[
\int_0^{2\pi} \frac{1}{(2+\sin \theta)^2}\,d\theta = \frac{4\pi}{3\sqrt{3}}
\]
(which is Supplementary Exercise 4 on page 71). Since there is a double pole inside the unit circle, you will need to apply either the method shown in the gray box on page 65 of the textbook or Cauchy's integral formula for the first derivative.
If you look back at the formulas you previously derived for stereographic projection, you will see that these formulas supply a parametrization of the unit circle by rational functions. Namely, if
\[
\cos\theta = \frac{2t}{t^2 +1}
\]
and
\[
\sin\theta = \frac{t^2-1}{t^2+1},
\]
then the point \( (\cos\theta,\sin\theta)\) traverses the unit circle when the variable \(t\) traverses the real axis. Show that under this change of variable,
\[
d\theta= \frac{2}{t^2+1}\,dt.
\]
Use the indicated change of variable to show that
\[
\int_0^{2\pi} \frac{1}{(2+\sin \theta)^2}\,d\theta =
\int_{-\infty}^\infty \frac{2(t^2+1)} {(3t^2 +1)^2}\,dt.
\]
(Thus the trigonometric integral that you computed above can be transformed into the integral of a rational function. This technique used to be taught in calculus courses.)
Use the method of Section 9B to show directly that
\[
\int_{-\infty}^\infty \frac{2(t^2+1)} {(3t^2 +1)^2}\,dt = \frac{4\pi}{3\sqrt{3}}.
\]
Tuesday, October 30
We discussed singularities, Riemann's theorem on removable singularities, and the residue theorem.
Here is the assignment to hand in next time.
Supplementary Exercise 1 on page 62, which asks for a discussion of the singularity of \(z/(\sin z)^2\) at the origin.
Supplementary Exercise 5 on page 62, which asks for a discussion of the singularity of
\[
\frac{1}{z} \int_0^z e^{-t^2}\,dt
\]
at the origin.
Discuss the singularity at the origin of
\[
\dfrac{1} {1+ \dfrac{1}{1+ \dfrac{1}{1+\dfrac{1}{z}}} }.
\]
Sunday, October 28
I posted grades in eLearning
for the second examination.
Thursday, October 25
The second exam was given. There is no assignment to hand in next time, but you can start reading Section 8 of the textbook, and you can read my solutions to the exam.
Reminder: The second examination is scheduled for Thursday, October 25.
Thursday, October 18
In class, we discussed zeros of holomorphic functions.
The assignment for next time (not to hand in, since the second
exam is on Thursday, October 25) is to decide which of the
problems on the August
2011 qualifying exam can be solved using tools that we have
learned so far in the course.
Tuesday, October 16
In class, we revisited some of the problems from last time and discussed sufficient conditions for interchanging the order of integrals and for differentiating under the integral sign.
Here is the assignment to hand in next time. Recall from the end of class today that a function \( f\colon (0,\infty) \to (0,\infty)\) is called logarithmically convex if \(\ln f\) is convex, that is,
if \( (\ln f)''\ge 0\), or equivalently \(ff''\ge (f')^2\).
Give a concrete example of a function that is convex but not logarithmically convex.
Show that (as mentioned in class) the gamma function is logarithmically convex on the positive real axis.
Suggestion: recall the Cauchy–Schwarz inequality for real integrals, which in schematic form says that
\[
\left| \int fg \right|^2 \le \int f^2 \int g^2.
\]
Thursday, October 11
In class, we started off on a tangent by discussing the strangeness of high-dimensional Euclidean space. You can read more about the example of the growing sphere. Returning to one-dimensional complex analysis,
we worked in groups on Supplementary Exercises from pages 52–53.
The assignment for next time is to read Section 7H in the textbook (about the gamma function) and to solve the following problems. Notice that the complex exponential \(t^{z-1}\) appearing in the definition of the gamma function has a canonical meaning when \(t\gt 0\): since \(t^{x-1} = \exp\{(x-1)\ln(t)\}\) when \(x\) is real, it is natural to interpret \(t^{z-1}\) as \(\exp\{(z-1)\ln(t)\}\) when \(z\) is complex.
When you enter (-1/2)! at WolframAlpha, you get the response \(\sqrt{\pi}\). Is this response compatible with what you know about the gamma function?
Supplementary Exercise 3 on page 59, which asks you to show that the second-order derivative \(\Gamma''(x)\) is positive when \(x\) is a positive real number; and, consequently, the graph of \(\Gamma(x)\) for positive \(x\) has a single minimum point.
This problem is about the behavior of \(\Gamma(z)\) when \(z\) is far away from the real axis. The issue is whether or not
\[
\lim_{y\to\infty} \Gamma(x+iy)=0.
\]
As mentioned in the textbook, the reciprocal function \(1/\Gamma(z)\) is an entire function. A student argues that the function \(1/\Gamma(z)\) must be unbounded by Liouville's theorem and therefore \(\Gamma(x+iy)\to 0 \) when \( y \to \infty\). Criticize this argument.
Recalling that \(|\sin z|\to\infty\) when \(\Im z\to\infty\), a student claims that the equation
\[
\Gamma(z) \Gamma(1-z)=\frac{\pi}{\sin \pi z}
\]
justifies the conclusion that \(\Gamma(x+iy)\to 0 \) when \( y \to \infty\). Criticize this argument.
Is it in fact true that \(\Gamma(z) \to 0\) when \(\Im z \to \infty\)? Why or why not?
Tuesday, October 9
We continued the discussion of consequences of Cauchy's integral formula. Topics included Cauchy's estimates for derivatives, the proposition of Exercise 7.7 (an entire function that grows no faster than some polynomial is a polynomial), and the proposition that uniform convergence preserves holomorphicity.
The assignment to hand in next time is Supplementary Exercise 2 on page 50, which asks why the function \(z^{1/2}\) does not violate Exercise 7.7; and Supplementary Exercise 3 on page 52, which asks if the sequence \(\{[4x(1-x)]^n\}\) of real-valued functions converges uniformly on the open interval where \(0\lt x\lt 1\).
Thursday, October 4
In class, we discussed Cauchy's integral formula for derivatives, and we worked in groups on evaluating some integrals by using Cauchy's formula together with some trickery (power series expansions, partial fractions, and a change of variables).
The assignment to hand in next time consists of the following three variations on Cauchy's integral formula.
Show that if \(D\) is a disk with center \(z_0\) and radius \(R\), and \(f\) is holomorphic in a neighborhood of the closure of \(D\), then
\[
f(z_0) = \frac{1}{\pi R^2} \iint_D f(z)\,dx\,dy
\]
(where, as usual, \(z=x+iy\), and \(dx\,dy\) is the standard area element). This formula is a two-dimensional version of the mean-value property of holomorphic functions.
Show that if \(w\) is an arbitrary point in the open unit disk \(D\) (that is, the disk with center \(0\) and radius \(1\)), and \(f\) is holomorphic in a neighborhood of the closure of \(D\), then
\[
f(w) = \frac{1}{2\pi} \int_{\partial D} \frac{f(z)}{1-\overline{z}w} \,|dz|
\]
(where \(|dz|\) represents the element of arc length, as mentioned on page 34 of the textbook; on the boundary of the unit disk, \(|dz|\) and \(d\theta\) are the same).
This formula is the simplest case of an integral representation
associated with the name of the famous Hungarian–American
mathematician Gábor Szegő
(1895–1985). When \(w=0\), this formula reduces to the mean-value property of holomorphic functions. You can view the formula as saying that the value of a holomorphic function at a general point is a certain weighted average of the boundary values.
Show that in the same setting as the preceding problem,
\[
f(w)=\frac{1}{\pi} \iint_D \frac{f(z)}{(1-\overline{z}w)^2}\,dx\,dy.
\]
This integral representation is the simplest case of a formula
associated with the name of the famous Polish–American
mathematician Stefan Bergman (1895–1977). Here the weighted averaging takes place on the whole region rather than on the boundary.
Tuesday, October 2
We discussed some consequences of Cauchy's integral formula: the mean-value property of holomorphic functions, the infinite differentiability of holomorphic functions, Cauchy's inequality for the first derivative, and Liouville's theorem.
Here is the assignment to hand in next time.
We observed in class that the average of a holomorphic function around the boundary of a disk equals the value of the function at the center of the disk. Are there any non-holomorphic functions for which this averaging property holds?
If you average a holomorphic function around an ellipse, do you get the value of the function at the center of the ellipse?
Sunday, September 30
I posted grades in eLearning
for the first examination. I scaled the scores to make the
highest-scoring student have a grade of 100.
Thursday, September 27
The first exam was given. There is no assignment to hand in next time, but you can start reading Chapter 2 of the textbook, and you can read my solutions to the exam.
Tuesday, September 25
By way of review for the exam to be given on Thursday, we worked in groups on some problems from Chapter 1.
Thursday, September 20
In class, we followed up on \(\int_0^{1+i} \overline{z}\,dz\), observing that the real part is path independent. Then we looked at a sketch of a proof of Cauchy's integral formula with remainder (for real-differentiable functions that are not necessarily holomorphic).
The first examination, scheduled for next Thursday (September 27), covers Chapter 1 of the textbook. The assignment for Tuesday, September 25 (not to hand in) is to finish reading the chapter and to identify in each of the six sections of the chapter the hardest exercise in that section.
Tuesday, September 18
In class, we discussed Green's theorem, Cauchy's integral theorem, and Morera's theorem.
Here is the assignment to hand in next time. Some browsers have a glitch in the display of mathematics, so I mention that the integrands in all three problems involve the conjugate of \(z\).
Show that if \(C\) is the unit circle oriented in the usual counterclockwise direction, then
\[
\int_C \overline{z}\,dz = 2\pi i.
\]
Suggestion: The unit circle—the circle of radius \(1\) centered at the origin—can be parametrized conveniently via \(\exp(i\theta)\), where \(\theta\) varies from \(0\) to \(2\pi\).
Show that if \(C\) is the unit circle, as in the preceding problem, then
\[
\int_C \overline{z^2}\,dz = 0
\]
(even though the function \( \overline{z^2}\) is not holomorphic).
Can you find a region \(R\) for which \(\int_{\partial R} \overline{z}\,dz =0\)?
Thursday, September 13
In class, we discussed several convergence tests for infinite series involving monotonicity: Cauchy's condensation test, Dirichlet's test, and Abel's test; we discussed Stirling's formula; and we worked on some problems about determining the radius of convergence of a power series.
Here is the assignment to hand in next time.
Supplementary Exercise 1 parts (g) and (h) on
pages 21–22, which ask for the (open) disk of convergence of
the power series \(\displaystyle \sum_{n=1}^\infty \frac{(n!)^2}
{(2n)!} z^n\) (part g) and \(\displaystyle \sum_{n=1}^\infty
n^{-1/n} z^{n^2}\) (part h).
Supplementary Exercise 1 parts (a) and (b) on
page 23, which ask whether the series \(\displaystyle
\sum_{n=1}^\infty \frac{z^n}{n^2}\) (part a) and
\(\displaystyle \sum_{n=1}^\infty [nz^n-(n+1)z^{n+1}]\) (part b) converge
uniformly on the open unit disk.
Remarks: Typically one asks about uniform convergence on closed sets (especially compact sets), but the concept of uniform convergence makes perfectly good sense on any kind of set. Notice that the series in part b is not a power series as it stands, for consecutive terms interact with each other. The terms could be regrouped to form a power series, but rearranging the terms of an infinite series can change its behavior if the series does not converge absolutely. In other words, exercise caution.
Supplementary Exercise 5 on page 29, which asks
whether the function \(\tan z\) is ever equal to \(i\).
(Of course, more than a one-word answer is expected!)
Tuesday, September 11
We observed that inversion is an isometry of the extended complex numbers with respect to the spherical distance, and we looked at Cauchy's treatment of the root test (the \(\lim\sup\) version) and the ratio test in his 1821 Cours d'analyse.
The assignment to hand in next time is the following problem (intended to get the wheels turning regarding the machinery of limits). Suppose that \(\{a_n\}_{n=1}^\infty\) is a sequence of positive real numbers. Show that if \(\displaystyle \lim_{n\to\infty} \frac{a_{n+1}}{a_n}\) exists and equals \(A\), then \(\displaystyle\lim_{n\to\infty} a_n^{1/n}\) necessarily exists and equals \(A\); concerning the converse, give a concrete example to demonstrate that existence of \(\displaystyle\lim_{n\to\infty} a_n^{1/n} \) does not necessarily imply existence of \(\displaystyle\lim_{n\to\infty} \dfrac{a_{n+1}}{a_n}\). (The upshot is that the root test is more discerning than the ratio test.)
Thursday, September 6
We revisited the Cauchy–Riemann equations and the definition of the complex derivative. Then we worked in groups on computing formulas for the two common models of stereographic projection: a sphere tangent to the plane and with north pole at the point \( (0,0,1)\); and a sphere with center at \( (0,0,0)\) and north pole at the point \( (0,0,1)\).
The assignment to hand in next time is to complete this task for whichever model you were working on in class [that is, find formulas for coordinates \(x\) and \(y\) in the complex plane in terms of the coordinates \(x_1\), \(x_2\), and \(x_3\) of the corresponding point on the sphere, and find formulas for \(x_1\), \(x_2\), and \(x_3\) in terms of \(x\) and \(y\)], and additionally determine a formula for the spherical distance between two complex numbers \(z\) and \(w\). In other words, determine the straight-line distance in \(\Bbb{R}^3\) between the spherical images of \(z\) and \(w\). The answer should be some expression involving \(|z-w|\) (the ordinary distance between \(z\) and \(w\)) and \(|z|\) and \(|w|\). Of course your answer will depend on which of the two models of stereographic projection you are using.
Tuesday, September 4
We discussed a geometric interpretation of the Cauchy–Riemann equations (the level sets of the real part of a holomorphic function are orthogonal to the level sets of the imaginary part of the function at points where the gradients are nonzero), and we worked out in groups the details of the geometry of the quadratic mapping that sends \(z\) to \(z^2\).
The assignment to hand in next time is Supplementary Exercise 1 on page 18, which asks for the Cauchy–Riemann equations when the dependent variable is expressed in polar coordinates and the independent variable is expressed in rectangular coordinates. In other words, if a function is written in the form \( R(x,y)(\cos\Phi(x,y) + i\sin\Phi(x,y))\), then the Cauchy–Riemann equations relate the partial derivatives \(\partial R/\partial x\), \(\partial \Phi/\partial y\), \(\partial R/\partial y\), and \(\partial\Phi/\partial x\). What are those relations?
Notice that Exercise 2.4 on page 16 addresses the same question but for other choices of coordinate systems (and Exercise 2.4 has a solution in the back of the book).
Thursday, August 30
We discussed the notion of complex differentiability and the Cauchy–Riemann equations.
The assignment for next time is to continue reading in the textbook (Section 2 of Chapter 1) and to solve Problem 10 from the August 2012 qualifying examination, which asks for a characterization of the set of all real harmonic functions \(u(x,y)\) with the property that the function \((x^2-y^2)u(x,y)\) is harmonic too.
Amplification: The goal is to obtain an explicit description of such functions. Your answer might be something like “all first-degree polynomials” or “all functions of the form \(a\sin(x)+b\cos(y)\), where \(a\) and \(b\) are arbitrary real numbers.”
(Neither of these examples is the correct answer.)
This problem is non-routine, so you should expect to make some initial progress and then get stuck. One possible way to get unstuck is to try to find as many examples as you can, and then see if you can prove that you have them all.
Tuesday, August 28
In class, we discussed various definitions and properties of the complex numbers. In particular, we saw that the complex numbers differ from the real numbers in several ways: the field of complex numbers is algebraically closed, the complex numbers cannot be disconnected by removing a single point, and the complex numbers cannot be equipped with a natural order relation. We ended by observing that the real-linear transformations of the complex numbers that correspond to complex-linear transformations are the ones obtained by composing rotations and dilations.
The standing assignment is to read the textbook. You should read at least pages 1–11 (Section 1 of Chapter 1) by next class.
Here is the assignment to hand in next class.
Supplementary exercise 2 on page 6, which asks for a geometric description of the set of complex numbers \(z\) described by each of the following relations.
\(|z+i|\lt 5\)
\(|2i-z|=|z+1+3i|\)
\(2\gt \Re z \gt -3\)
\(\Re[(3-4i)z]\gt 0\)
\( (\Im z)^2\le \Re z\)
\(2\Re z \lt |z|^2\)
Supplementary exercise 2 on page 8, which asks for the value of each of the following expressions both when “arg” means the principal value of the argument (an angle between \(-\pi\) and \(\pi\)) and when the argument is taken between \(0\) and \(2\pi\).
\(\arg z + \arg\overline{z}\)
\(\arg (z\pm \overline{z}\,)\)
\(\arg(z/\,\overline{z}\,)\)
\(\arg \overline{(z-\overline{z}\,)}\)
Friday, August 24
Welcome to Math 617. This site went live today. I will post regular updates during the semester.