Journal for Math 617 Fall 2010

December 16
I posted the scores for the final exam and the course letter grades at the TAMU eLearning site.
December 15
The final exam was given.
December 7
In class today, we compiled a list of eleven theorems to know for the final exam, sketched the solution of the Dirichlet problem for the unit disk, and did the written course evaluations for the mathematics department.
Please go to pica.tamu.edu to do the electronic evaluation for the university SLATE (or SRATE) program before midnight on Thursday, December 9.
The final exam takes place on December 15 (Wednesday), 1:00–3:00pm, in the usual classroom.
December 2
In class today, we discussed the analytic automorphism groups of the unit disk and the upper half-plane, and we started a discussion of the Dirichlet problem on the unit disk (to be completed next time, which is the last class meeting for the semester).
The assignment for next time (not to hand in) is to compile a list of the major theorems from the course.
November 30
In class today, we discussed the invariance of the cross ratio under Möbius transformations, the notion of points symmetric with respect to a circle, the subgroup of Möbius transformations that preserve the unit disk, and the Schwarz–Pick lemma.
November 29
Here is the assignment due on Thursday, December 2.
  1. Suppose \(f\) is an analytic function that maps the unit disk into (not necessarily onto) itself. If \(f(1/2)=0\), then how large can the value of the derivative \(|f'(1/2)|\) be?
  2. Find the most general Möbius transformation that maps the upper half-plane onto itself.
  3. Suppose that \(u\) is harmonic in a neighborhood of the closed unit disk. Derive the Poisson integral formula \[ u(r e^{i\theta}) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1-r^2}{1-2r\cos(\theta-t) +r^2} u(e^{it}) \,dt\] (where \(0\lt r\lt 1\)) by using the following method. You already know the mean-value property \[u(0)=\frac{1}{2\pi} \int_0^{2\pi} u(e^{it})\,dt.\] Apply this property to the harmonic function \(u\circ \psi_z\), where \(z=re^{i\theta}\) and \(\psi_z\) is the standard disk automorphism that interchanges \(0\) and \(z.\) Make a change of variables in the resulting integral to obtain Poisson’s formula.
November 23
Today was the day of the third exam, for which solutions are available.
November 18
In class today, we discussed some further properties of Möbius transformations and proved Riemann’s removable-singularities theorem.
The third exam takes place next class.
November 16
In class last Thursday, we discussed Rouché’s theorem. In class today, we discussed the notions of conformality, projective space, and linear fractional transformations.
The assignment for the coming week is to study for the third examination, which takes place on Tuesday, November 23. The exam covers Sections 4.1–4.5 in the textbook.
November 10
Here is the assignment due on Tuesday, November 16.
  1. Solve the following problem from the August 2008 qualifying examination in complex analysis:
    Let \(\alpha\) be a real number larger than \(1.\) Show that there are exactly three solutions of the following equation in the unit disk: \[\sin(z)=e^\alpha z^3.\]
  2. Suppose that \(p\) and \(q\) are polynomials, the degree of \(q\) is at least two more than the degree of \(p,\) and \(q\) has no zeroes on the real axis. Prove that \[\int_{-\infty}^\infty \frac{p(x)}{q(x)}\,dx\] equals \(2\pi i\) times the sum of the residues of the rational function \(p/q\) at the poles in the upper half-plane.
    Hint: Compute a path integral over a semi-circle of large radius \(R\) and take the limit as \(R\) tends to infinity.
  3. Solve the following problem from the January 2009 qualifying examination in complex analysis:
    Prove that when \(n\) is an integer greater than or equal to \(2\) and \(a\) is an arbitrary complex number, the polynomial \(1+z+az^n\) has at least one zero in the closed disk where \(|z|\le 2.\)
    Hint: Argue by contradiction, using the fact from algebra that the product of the roots of a monic polynomial equals plus or minus the constant term of the polynomial. (But notice that the given polynomial is not monic.)
November 9
In class today, we discussed the homology version of the residue theorem, the argument principle, and an example of approximately locating the zeroes of a polynomial.
November 8
Here is the assignment due on Thursday, November 11.
  1. Use the residue theorem to prove that \[ \int_0^\pi \frac{1}{1-2r\cos(\theta)+r^2}\,d\theta = \frac{\pi}{1-r^2} \] when \(0\lt r \lt 1.\)
  2. Evaluate the integral \[ \int_{|z|=1} \frac{100 z^{99} -2}{z^{100}-2z}\,dz.\]
  3. Suppose that \(f\) is analytic—except for isolated singularities—on a region \(\Omega\) that is symmetric with respect to the real axis, and \(f\) takes real values on the part of the real axis in \(\Omega\). Show that if \(f\) has a pole at a point \(z_0\) in \(\Omega\) with residue equal to \(b,\) then \(f\) has a pole at the complex-conjugate point \(\overline{z_0}\) with residue equal to the complex-conjugate value \(\overline{b}.\)
November 4
In class today we discussed the integral formula for coefficients of Laurent series, the notion of residue, the basic version of the residue theorem, and a concrete real integral that can be evaluated using residues.
November 3
Here is the assignment due on Tuesday, November 9.
  1. Let \(\gamma\) be the unit circle with its standard counterclockwise orientation, and let \(k\) be an integer (possibly negative). Evaluate the path integral \[ \int_\gamma z^k \exp(1/z) \,dz.\]
  2. Suppose \(f\) and \(g\) are functions with isolated singularities at a point \(z_0.\)
    1. Is the residue at \(z_0\) of the sum function \(f+g\) equal to the sum of the residue of \(f\) and the residue of \(g\)?
    2. Is the residue at \(z_0\) of the product function \(fg\) equal to the product of the residue of \(f\) and the residue of \(g\)?
  3. Determine the possible values of \[ \int_\gamma \tan(z)\,dz \] when \(\gamma\) is a simple, closed curve in the complex plane.
November 2
In class today, we discussed isolated singularities and Laurent series.
November 1
Here is the assignment due on Thursday, November 4.
  1. Discuss the singularities of the function \[ \frac{z \sin(z)}{1-\cos(z)}.\]
  2. Find both Laurent series in powers of \((z-2)\) and \(1/(z-2)\) for the function \[ \frac{z}{(z-1)(z-2)(z-3)}.\]
  3. Consider on \(D(0,1)\), the unit disk, the following analytic function: \[f(z)=\exp\left( \frac{1+z}{1-z} \right).\] Evaluate the radial limit \[ \lim_{r\to 1^{-}} |f(re^{i\theta})|\] as a function of the angle \(\theta.\)
October 31
I have posted grades for the second exam at the TAMU eLearning site.
October 28
In class on October 26, we reviewed for the exam, did three proofs of the fundamental theorem of algebra using complex analysis, and previewed coming attractions. Today was the day of the second exam, for which solutions are available. Next time we will start Chapter 4.
October 25
I updated the comments on Chapter 4.
Reminder: the second exam takes place in class on Thursday, October 28.
There is no assignment due on Tuesday, November 2.
October 22
In class yesterday, we extended the list of properties that are equivalent to simple connectivity of a planar domain, and we discussed stereographic projection as a method for creating models of the extended complex numbers.
The assignment for Tuesday, October 26 is to make a list of the main theorems from Chapter 3 and the last section of Chapter 2 (not to hand in). The second exam takes place on Thursday, October 28.
October 19
In class today, we did a proof of the homology version of Cauchy’s integral formula and then started compiling a list of properties of a domain in the plane that are equivalent to simple connectivity.
October 18
Here is the assignment due on Thursday, October 21.
  1. You know that \(\sin(z)\) is a real number when \(z\) is a real number (that is, a complex number with imaginary part equal to zero). Are there any other complex numbers \(z\) for which \(\sin(z)\) is real? If so, can you find all of them?
  2. Show that if \(\mathop{\textrm{Log}} \) denotes the principal branch of the logarithm, then \[ \mathop{\textrm{Log}} (1+z) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} z^n\] when \(|z|\lt 1\).
  3. Suppose \(\gamma\) is a closed path, not necessarily simple. Determine the possible values of the path integral \[ \int_\gamma \frac{1}{z^2 -1}\,dz.\]
October 15
In class yesterday, we discussed logarithms of functions and the concept of winding number, and we ended with the statement of the homology form of Cauchy’s theorem and Cauchy’s integral formula.
October 13
In class yesterday, we discussed complex logarithms.
Here is the assignment due on Tuesday, October 19.
  1. Find all complex numbers \(z\) such that
    1. \(\tan(z)=1\)
    2. \(\tan(z)=i\)
  2. Consider the exponential function \(\exp(z)\) on the right-hand half-plane where \(\mathop{\textrm{Re}}(z)\gt 0\). On the boundary of this region (that is, on the imaginary axis), the modulus of the exponential function is equal to \(1\), yet \(1\) is not the maximum of the modulus of the exponential function on the region. Explain why the maximum principle is not contradicted.
  3. You know that the exponential function \(\exp\) has the property that \[\exp(z+w)=\exp(z)\exp(w)\] for all complex numbers \(z\) and \(w\). Are there other entire functions \(f\) with the analogous property that \[f(z+w)=f(z)f(w)\] for all complex numbers \(z\) and \(w\)? Can you determine all such entire functions?
October 11
In class on Thursday, October 7, we discussed zeroes of analytic functions and the identity theorem; revisited the maximum principle; and proved the Schwarz lemma. In class tomorrow, we will discuss complex logarithms.
Here is the assignment due on Thursday, October 14.
  1. When \(z\) is a real number and \(-\pi/2\lt z\lt \pi/2,\) the modulus of the analytic function \(\cos(z)\) attains a maximal value of \(1\) (for \(z\) equal to \(0\)). But according to the maximum principle, a nonconstant analytic function should not attain its maximum modulus at an interior point of the domain. Resolve the apparent contradiction.
  2. Suppose \(f\) is analytic in a neighborhood of the closed unit disk. If \(f(0)=0,\) and \(|f(z)|\le |e^z|\) when \(|z|=1,\) then how large can \(|f(\ln 2)|\) be?
    [A complete solution should include both an upper bound and an example to show that the bound can be attained.]
  3. You know from a previous homework problem that there is no analytic square-root function defined in a neighborhood of \(0\). Show nonetheless that there exists an entire function \(f(z)\) such that \(f(z^2)=\cos(z)\) for every complex number \(z.\)
    [One would like to say that \(f(z)=\cos\sqrt{z},\) but the right-hand side does not quite make sense.]
October 6
Here is the assignment due on Tuesday, October 12.
  1. Prove that if \(f\) is an entire function such that \(\mathop{\textrm{Re}} f(z)\gt 0\) for all \(z,\) then \(f\) is constant.
  2. Prove that \( (1+\cos z)/2 = \cos^2(z/2)\) for every complex number \(z.\)
  3. Suppose that \(f\) is an analytic function on the unit disk, and \(|f(z)|\lt 1\) when \(|z|\lt 1.\) Prove that the derivative satisfies the following inequality: \[|f'(0)|\le 1.\] What can you say about the function \(f\) if equality holds in this inequality?
October 5
In class today, we discussed Cauchy’s estimates for derivatives, Liouville’s theorem, the mean-value property of analytic functions on disks, and the maximum principle for analytic functions.
October 4, second post
Here is the assignment due on Thursday, October 7.
Find the maximum on the closed unit disk \( \{\, z\in{\mathbb{C}}: |z|\le1\,\}\) of the modulus of each of the following analytic functions.
  1. \(z^2 + z+1\)
  2. \(\exp(z)\)
  3. \(\sin(z)\)
October 4
I have posted grades for the first exam at the TAMU eLearning site.
September 30
The first exam was given, and solutions are available.
September 28
In class today, we discussed uniform convergence of sequences of analytic functions, remarked on the Bieberbach Conjecture (now a theorem of de Branges) as an example of the general problem of relating properties of an analytic function to properties of its sequence of series coefficients, and reviewed briefly for the exam to be held next class.
September 23
In class today, we discussed the proof of Cauchy’s integral formula, the application to power series expansions of analytic functions, and Morera’s theorem.
The homework assignment due on Tuesday, September 28 is to read Section 2.3 (about the exponential and trigonometric functions) and to work the problems at the end of that section. But these problems are not to be turned in. (The authors' solutions to all the exercises in the textbook are available.)
Reminder: The first exam takes place in class on Thursday, September 30. The exam covers Chapter 1 and Sections 2.1–2.3 in the textbook.
September 22
I have posted comments on Homework 6.
September 21
In class today we revisited Goursat’s proof of Cauchy’s theorem, observing that the argument can be adapted to apply to a function analytic inside any simple, closed curve or even to a function analytic in the region between two simple, closed curves. Consequently, the value of a path integral does not change when the path is deformed within the region where the integrand is analytic. Then we looked at Cauchy’s integral formula that represents an analytic function in the region inside a simple, closed curve through an integral over the boundary curve. Differentiating under the integral sign provides a corresponding integral formula for the derivative of an analytic function.
September 20
Here are the homework problems due on Thursday, September 23.
  1. Suppose \[\gamma(t) = \cos^{2}(t) + i \sin^{2}(t),\] where \(0\le t\le 2\pi.\) Evaluate the path integral \[\int_{\gamma} \frac{1}{z}\,dz.\]
  2. Suppose the curve \(\gamma\) represents a circle of radius \(2\) centered at the origin: namely, \(\gamma(t)=2\exp(it),\) where \(0\le t\le 2\pi\). Evaluate the path integral \[ \int_{\gamma} \frac{z^{617}}{z^{2010} -1}\,dz.\] Hint: What can you say about the corresponding integral over a very large circle?
  3. Suppose that \(f\) is analytic in the open disk \(D(z_{0},r),\) and \(M\) is a positive real number such that \[|f(z)|\lt M\] for all \(z\) in this disk. Prove that \[|f'(z_{0})| \le M/r.\]
September 16
In class today, we discussed the basic principles concerning convergence of infinite series of complex numbers and looked at some examples. In particular, we discussed the radius of convergence \(R\) of a power series \(\sum_{n=1}^\infty a_n z^n\) and the theorem (which goes back to Cauchy) that \[ \frac{1}{R} = \limsup_{n\to \infty} |a_n|^{1/n}.\]
September 15
Here are the homework problems due on Tuesday, September 21.
  1. Find an example of a sequence \(\{a_n\}\) of complex numbers such that the series \[\sum_{n=1}^\infty a_n\] converges (conditionally), yet the series \[\sum_{n=1}^\infty a_n^3\] diverges.
  2. Determine the set of complex numbers \(z\) for which the series \[\sum_{n=1}^\infty (1-z^2)^n\] converges. (This series is not a power series! The convergence region is not a disk.)
  3. Prove that the power series \[\sum_{n=1}^\infty a_n z^n\] and the formally differentiated power series \[\sum_{n=1}^\infty n a_n z^{n-1}\] have the same radius of convergence.
September 14
In class today, we discussed two fundamental inequalities for integrals: \[\left| \int_a^b f(t)\,dt\right| \le \int_a^b |f(t)|\,dt\] for a complex-valued function \(f\) on a real interval \( [a,b],\) and \[ \left|\int_\gamma f(z)\,dz \right| \le ML\] where \(M\) is an upper bound for the modulus of \(|f|\) on the curve \(\gamma\) and \(L\) is the length of the curve. Also we covered Goursat’s proof of Cauchy’s theorem for the special case of a rectangle.
September 13
Here are the homework problems due on Thursday, September 16. In these problems, let \(r\) be a positive real number, and let \[\gamma_r\colon [0,\pi] \to \mathbb{C}\] denote a semicircular path of radius \(r\), that is, \[\gamma_r(t)=r(\cos(t)+i\sin(t)).\] Notice that this path is not a closed path.
  1. Evaluate the path integral \(\int_{\gamma_r} z\,dz.\)
  2. By estimating the modulus of the integral, show that \[\lim_{r\to 0} \int_{\gamma_r} \frac{z^2+1}{z^4+1}\,dz =0.\]
  3. By estimating the modulus of the integral, show that \[\lim_{r\to \infty} \int_{\gamma_r} \frac{z^2+1}{z^4+1}\,dz =0.\]
[Remark: These problems illustrate that \(0\) can arise as an answer for a variety of reasons.]
September 10
In class yesterday we discussed the notion of path integrals (“line integrals”) in the complex plane, worked some examples, and proved two propositions about the integral of a function over a simple closed curve being equal to zero: this conclusion holds if the function has a primitive (an anti-derivative) on an open set containing the image of the curve, and it holds too if the function is analytic on an open set that contains both the curve and the region inside the curve (Cauchy’s theorem). We did Riemann’s proof of Cauchy’s theorem (using Green’s theorem).
I have posted comments and a solution for the second problem (the hard problem) on the homework that you turned in yesterday.
September 8, second post
Here is the homework assignment due next Tuesday, September 14.
  1. One of the problems that Cauchy studied while developing his theory of integrals “between imaginary limits” is \[\int_{-1}^1 \frac{1}{z^2}\,dz.\] You know from real calculus that if the integration path is a segment of the real axis, then this integral diverges because of the fast rate of blow-up of the integrand at the origin. What can you say about the integral if the integration path lies in the complex plane and avoids the origin?
  2. In real calculus, there is a mean-value theorem for integrals stating that if \(f\) is a continuous real-valued function on an interval \([a,b]\) of the real axis, then there is a point \(c\) in the interval such that \[\int_a^b f(t)\,dt=(b-a)f(c).\] The proof uses the order relation on the real numbers, so you might suspect that the corresponding statement fails for integrals of complex-valued functions. Show, indeed, that there is no such mean-value formula for the integral of the complex-valued function \(\exp(it)\) on the interval \([0,2\pi]\) of the real axis.
  3. Suppose that \(\gamma\) is a simple, closed, continuously differentiable curve in the complex plane. You know by Cauchy’s theorem that \[\int_\gamma z\,dz=0.\] What can you say about \[\int_\gamma \overline{z}\,dz?\] (The integrand \(z\) now has been replaced by the conjugate of \(z\), a non-analytic function.)
September 8
In class yesterday, we discussed three formulations of complex differentiability, defined harmonic functions, and did a computation using a path integral (“line integral”) to determine a harmonic function \(v\) such that \(u+iv\) is analytic when \(u(x,y)=e^x \cos(y)\).
I did not mention the terminology in class, but perhaps you read on page 9 of Chapter 1 of the textbook that such a function \(v\) is called a harmonic conjugate of \(u\) (“a”, not “the”, because one can always add a constant to \(v\) to get another harmonic conjugate). Thus the task in the third problem in the homework is to show that \(u+iv\) is an analytic function. Incidentally, in the statement of that problem, I am using the notation “log” to denote the natural logarithm function (to base \(e\)), as is traditional in advanced mathematics. Our textbook (in the notation introduced in Chapter 3) would write either “ln” or “Log” (with a capital letter).
Notice also that the relation of being conjugate functions is not a symmetric relation. If \(v\) is a harmonic conjugate of \(u\), then a harmonic conjugate of \(v\) is not \(u\) but instead \(-u\).
The terminology “conjugate functions” was introduced by the Irish scientist William Rowan Hamilton (1805–1865) in his 1835 Theory of conjugate functions, or algebraic couples. The concept seems to have been popularized by the Scottish physicist James Clerk Maxwell (1831–1879) by its inclusion in his 1873 Treatise on Electricity and Magnetism (volume 1, article 183, page 227).
September 6
Here are the homework exercises due on Thursday, September 9.
  1. Show that for every nonzero complex number \(w\), there are infinitely many complex numbers \(z\) such that \(\exp(z)=w\).
  2. Show that there are infinitely many complex numbers \(z\) such that \(\exp(z)=z\). (This problem is more difficult than the preceding one.)
  3. Show that if \(u(x,y)=\log(x^2+y^2)\) and \(v(x,y)=2\arctan(y/x)\), then the functions \(u\) and \(v\) are harmonic conjugates on the right-hand half-plane where \(x\gt 0\). (The restriction to the right-hand half-plane ensures that the fraction \(y/x\) makes sense.)
September 2
After reviewing the notion of real-differentiability of a function from \({\mathbb{R}}^2\) to \({\mathbb{R}}^2\), we saw that requiring the real Jacobian matrix to correspond to a complex-linear transformation of \(\mathbb{C}\) leads to the Cauchy–Riemann equations, which are therefore necessary conditions for complex differentiability. The example \(z^4/|z^4|\) shows that the Cauchy–Riemann equations at a single point are not a sufficient condition for complex differentiability at the point. On the other hand, the Looman–Menshov theorem [which we shall not prove] says that for continuous functions, the Cauchy–Riemann equations on an open set are a sufficient condition for complex differentiability on the set.
The following homework problems were assigned (due next class, Tuesday, September 7).
  1. Show that if a function \(f\) is differentiable in the complex sense on a connected open subset of \(\mathbb{C}\), and \(f\) takes only real values, then \(f\) reduces to a constant function.
  2. Suppose \(f\) is differentiable in the complex sense on some open set. Let \(u\) denote the real part of \(f\), and let \(v\) denote the imaginary part of \(f\). Explain why the level curves of \(u\) and the level curves of \(v\) must be mutually orthogonal. (The level curves are defined by \(u=\text{constant}\) and \(v=\text{constant}\).)
  3. Show that there is no function \(f\) that is differentiable in the complex sense in a neighborhood of the origin and that satisfies the identity \(( f(z))^2 \equiv z\). (In other words, there is no complex-differentiable square-root function in a neighborhood of the origin.)
August 31
We discussed geometric and algebraic characterizations of the complex numbers, the standard distance on the complex numbers, limits of sequences of complex numbers, and terminology from the theory of metric spaces.
The following homework problems were assigned (due next class). Next time we will start discussing functions of a complex variable.
August 26
Welcome to Math 617. I will be regularly updating this page with homework assignments, brief summaries of class activities, and other information.
Reading the textbook is essential. If you encounter unclear points in the book, please let me know, and I will update my comments on the textbook.

Harold P. Boas