# Math 4180: Complex Analysis

Spring 2018
Malott 224, Tuesday & Thursday 11:40-12:55

Course Information

Instructor: Professor Robert Strichartz
• Office: 563 Malott
• Office Hours: Monday, Wednesday 10:30 - 11:30 AM
• If the weather is nice, office hours may be held in the flower garden between the AD White House and the Big Red Barn.
• Email: str(at)math.cornell.edu
• Office Phone: 255-3509
TA: Andres Fernandez
• Office Hours: Wednesday 2-4 PM in Malott 218
• Email: ajf254(at)cornell.edu

Text: Complex Analysis by Theodore Gamelin. Free download available from the library.

Interview: Please come for a short interview during the first week. There will be a sign-up sheet at the first class.

Course description: This is an introductory course in complex analysis, with an emphasis on understanding the proofs of the basic theorems. (By contrast, Math 4220, offered in the Fall semester, covers roughly the same material, but with an emphasis on applications). We will cover most of the first two parts of "Complex Analysis" by Theodore Gamelin.

• Class participation (25%)
• Written homework - weekly assignments due in class on Thursdays (25%)
• Midterm - in class March 15 (20%)
• Final exam - May 14, 9:00 - 11:30 AM (30%)

"A Cornell student's submission of work for academic credit indicates that the work is the student's own. All outside assistance should be acknowledged, and the student's academic position truthfully reported at all times" ... Cornell University Code of Academic Integrity

How the course will work: For each class there will be an assigned reading from the text and a set of discussion questions. You will do the reading and think about the questions before the class. Most of the class time will be devoted to discussing the questions. We will also discuss questions that you bring up, and may go over some homework problems. It is expected that everyone will participate in the discussion. Approximately 25% of your grade will be based on your enthusiastic participation. The goal is to get each student to understand the material through a learning process that is messy, challenging, individualized, frightening, thrilling, and ultimately transcendental.

Homework: You are allowed to work together with other students on HW, provided you write up the solutions on your own. Please write the names of the students you worked with on the top of your HW paper

Schedule

Hw, due Problems

#1, Feb 8

I.1 - 7
I.2 - 2(a, c)
I.3 - 5
I.4 - 1(a), 2(a)
1.5 - 1(d), 2(b)
I.6 - 1(c), 2(b)
I.7 - 3(b), 4
I.8 - 1(c)
II.1 - 6(a, b, c), 7, 16

#2, Feb 15

II.2 - 1(c, d), 3
II.3 - 3, 4, 6, 7(b)
II.4 - 1(a), 6, 10
II.5 - 1(b), 2, 5, 6

#3, Mar 1

II.6 - 1(a), 2, 3(a)
II.7 - 1(b), 3, 7
III.1 - 2, 5
III.2 - 3
III.3 - 2, 5
III.4 - 3
III.5 - 3, 4, 10

#4, Mar 8

IV.1 - 3(b), 5
IV.2 - 4, 5
IV.3 - 4, 6
IV.4 - 3, 4
IV.5 - 3
IV.6 - 3

#5, Mar 22

V.1 - 4, 5
V.2 - 7, 10
V.3 - 4, 7
V.4 - 9, 11
V.5 - 2, 3
V.7 - 1(e, f, g), 7, 9, 11
V.8 - 4, 5, 8

#6, Mar 29

VI.1 - 1(b, c), 2(b, c)
VI.2 - 1(b, d), 5, 6, 9, 10, 12
VI.3 - 1(b, d), 4(a, b)
VI.4 - 1(a, c)
VI.5 - 1, 3

#7, Apr 12

VI.6 - 6, 7, 8
VII.1 - 1(c, d), 2(c), 3(b,c)
VII.2 - 4, 8, 12
VII.3 - 2, 5, 6

#8, Apr 19

VII.4 - 1, 5
VII.5 - 2, 4
VII.6 - 4, 6
VIII.1 - 1, 3, 4, 7
VIII.2 - 5
VIII.3 - 2

#9, Apr 26

VIII.4 - 3
VIII.5 - 4, 6, 9, 11
VIII.6 - 2, 3
VIII.8 - 7, 8 (in this exercise $\varphi$ is also assumed to be 1-1)

# Tues. Jan 30

Read: I - 1, 2, 3
Questions:
1. What is the relationship between complex polynomials and polynomials in $\mathbb{R}^2$ with complex coefficients?
2. Is there any algebraic way to distinguish $i$ and $-i$?
3. If you need to add complex numbers, is their polar coordinate representation helpful?
4. The book says (2.2) is "convenient". What other words would you use to describe it?
5. What point in the complex plane corresponds to the south pole on the sphere under stereographic projection? What about the equator of the sphere? What does this tell you about the northern and southern hemisphere?
6. Can you say anything about centers of circles under stereographic projection?

# Thurs. Feb 1

Read: I - 4, 5, 6
Questions:
1. How would you describe the function $z^2$ in terms of polar coordinates? What about the function $z^{\frac{1}{2}}$
2. What goes on at the points lying over 0 on the Riemann surface of $w^{\frac{1}{2}}$
3. Is the function $e^{iz}$ periodic? If so, what is the period?
4. Describe a strip on which $z \mapsto e^{z}$ is one-to-one and onto the punctured plane?
5. How can you use the logarithm and exponential function to understand the square and square root function?

# Tues. Feb 6

Read: I - 7, 8. II - 1
Questions:
1. How many values of $z^{\alpha}$ are there if $\alpha$ is a rational real number?
2. Can you analyze the behavior of ${(z - 1/z)}^{\frac{1}{2}}$ by taking the logarithm?
3. Is $\tan{z}$ periodic?
4. In trying to understand the convergence of complex objects, is it equivalent to understand the convergence of the real and imaginary parts?
5. What is a Cauchy sequence, and why is this a significant idea?
6. Is the annulus $1 < |z| < 2$ a domain?
7. What is the relationship between closed sets and open sets?

# Thurs. Feb 8

Read: II - 2, 3
Questions:
1. A polynomial in 2 variables has the form $p(z_1, z_2) = \sum_{j = 0}^{n} \sum_{k = 0}^{m} a_{jk} {z_1}^{j} {z_2}^{k}$. When is $f(z) = p(z, \bar{z}))$ analytic?
2. Why is the definition of the derivative in the complex case more subtle than in the real case?
3. How would you express the Cauchy-Riemann equation in terms of the matrix $\begin{bmatrix} \frac{\partial{u}}{\partial{x}} & \frac{\partial{u}}{\partial{y}} \\ \frac{\partial{v}}{\partial{x}} & \frac{\partial{v}}{\partial{y}} \end{bmatrix}$
4. How does the proof that Cauchy-Riemann implies analytic relate to the theory of directional derivatives of functions on R^2
5. If D is an open set that is not connected, what does $f'(z) = 0$ on $D$ imply about $f$?
6. How does the identity $\frac{\partial}{\partial{\bar{z}}} f = 0$ relate to the Cauchy-Riemann equations?

# Tues. Feb 13

Read: II - 4, 5
Questions:
1. Give a geometric description of the Jacobian matrix $J_f$ of an analytic function. How does this relate to the inverse function theorem (p. 51)?
2. If $f : D -> D'$ is an onto analytic function with $f'(z) \neq 0$ on $D$, does this imply the existence of an inverse function $f^{-1} : D' -> D?$
3. If $f$ is analytic on $D$ and $f'(z_0) = 0$ for some $z_0$ in $D$, is it possible that $f$ nevertheless has a local analytic inverse in a neighborhood of $z_0$?
4. What justifies the interchange of the order of the derivatives in the sentence following (5.2)?
5. Does an analytic function $f$ satisfy $\Delta(f) = 0$?

# Thurs. Feb 15

Read: II - 6, 7
Questions:
1. If $f$ is analytic and $f'(z_0) \neq 0$, what does multiplication by $f'(z_0)$ do to complex numbers viewed as points in the plane?
2. In the definition of conformal mapping, why is it necessary to add the condition that the mapping be one-to-one? Doesn't this follow from $f'(z) \neq 0$?
3. In the figure on p. 61, what is going on at the origin?
4. Instead of requiring the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ to have nonzero determinant, suppose we require the determinant to equal 1. Does that leave out any fractional linear transformation? Does that eliminate all redundancy in the representation?
5. Is every fractional linear transformation a conformal map?
6. What does inversion do to the unit circle about the origin?

# Thurs. Feb 22

Read: III - 1, 2, 3
Questions:
1. Is a piecewise smooth path necessarily a smooth path?
2. In Green's Theorem, can you break up 1.4 into two equalities that are added together?
3. What determines the directions of the arrows in the figure on p. 73?
4. Can you translate the discussion of exact and closed differential forms into the language of gradient and curl?
5. Is there a converse to the lemma on p. 83?
6. Is the existence of a harmonic conjugate valid locally on any domain?

# Tues. Feb 27

Read: III - 4, 5 and IV - 1
Questions:
1. What does the mean value theorem for harmonic functions tell you about analytic functions?
2. Is the value of a harmonic function at the center of a square equal to the mean value on the boundary of the square?
3. What is "strict" about the Strict Maximum Principle?
4. What does applying the Strict Maximum Principle to $-w(z)$ tell you?
5. Would you define the complex integral $\oint_{\delta} h(z) dz$ directly as the limit of (1.2) without breaking it into the sum in (1.1)?
6. Do you expect that the ML estimate is typically sharp?
7. Is it surprising that the integral of $(z - z_0)^m$ on the bottom of p. 103 is independent of m for $m \neq -1$?

# Thurs. Mar 1

Read: IV - 2, 3, 4
Questions:
1. Does Part I of the FTC on p. 109 require that $D$ be starshaped?
2. Is there any analog of Cauchy's Theorem (p. 110) in real calculus?
3. If $f$ is analytic in the annulus $r < |z| < R$ how does $\oint_{|z| = s} f(z) dz$ vary with $s$ for $r < s < R$?
4. Is Cauchy's Theorem a special case of the Cauchy Integral Formula?
5. If we differentiate (4.1) we get (4.2), but how do we know the derivatives exist?

# Tues. Mar 6

Read: IV - 5, 6, 7
Questions:
1. What is $\lim_{m \rightarrow \infty} \frac{m!}{p^m}?$
2. What does Liouiville's Theorem imply about a bounded anlytic function on $\mathbb{C} - \{0\}$?
3. What does the theorem on the bottom of p. 121 tell you about gluing together analytic functions on $D \cap \{\text{Im} z > 0\}$ and on $D \cap \{\text{Im}z < 0\}$? Would it be any different if you replaced $\text{Im}z$ by $\text{Re}z$?
4. Is there a direct argument to show that the existence of the derivative at every point in $D$ implies that the derivative is continuous?
5. Does the existence of the derivative at a point of $z_0$ imply that the function is continuous at $z_0$?

# Thurs. Mar 8

Read: V - 1, 2, 3
Questions:
1. Can we substitute $z = -1$ into the identity $\sum_{k = 0}^{\infty} z^k = \frac{1}{1-z}?$
2. Is it true that $\frac{1}{1-z} - \sum_{k=0}^{n} z^k = \sum_{k = n+1}^{\infty} z^k$ for $|z| < 1$? Could we use this to deduce the estimate at the end of section V.1?
3. Why does the book say ''usually" for the formula for $\epsilon_j$ in the middle of p. 134?
4. How does the description of the example on the top of p. 136 illustrate the maxim ''If you don't like the answer, then change the question"?
5. Is the analog of the theorem on the bottom of p. 136 valid in real analysis?
6. Is the analog of the theorem on p. 138 valid in real analysis? What about the other theorems in section V.3?

# Tues. Mar 13

Read: V - 4, 5, 6
Questions:
1. Can you obtain Euler's formula $e^{iz} = \cos{z} + i \sin{z}$ from the power series on p. 145?
2. Does $f(z) = g(z)$ in $|z| < c$, where $f$ and $g$ are analytic in $|z| < r$ imply $f(z) = g(z)$ in $|z| < r$?
3. What is the radius of convergence of $f(z) = \frac{1}{1+ z^2}$ about a point $z_0$?
4. Is every entire function analytic at $z = \infty$?
5. What does the $\ldots$ in (6.1) really mean? In other words, if $a_m b_n$ appears in (6.1), what can you say about $m$ and $n$?
6. In the example on p. 153, would it be easier to compute the derivatives of $\tan{z}$ using the quotient rule?

# Thurs. Mar 15

In class midterm exam covering material up to section V.6.

# Tues. Mar 20

Read: V - 7, 8
Questions:
1. Can there be zeroes of infinite order of an analytic function? What about in real calculus?
2. If $f$ is an analytic function on an open set, do its zeroes have to be isolated?
3. Compare the Uniqueness Principle with the corollary on the top of p. 146.
4. Do you expect that the inequality (8.1) is usually an equality?
5. To have an analytic continuation along $\gamma$ is it necessary to have power series about every point $\gamma(t)$, or is it sufficient to have a power series about "a lot of points $\gamma(t)$"?
6. Can you have an analytic function in the disc $|z| < 1$ that has no analytic continuation outside that disc?

# Thurs. Mar 22

Read: VI - 1, 2
Questions:
1. The uniqueness of the Laurent decomposition requires $f_1(z)$ to vanish at $\infty$. What happens if we drop this requirement?
2. What is the relationship of the region where the full Laurent series $\sum_{-\infty}^{\infty} a_n(z - z_0)^n$ converges and the regions where the half Laurent series $\sum_{-\infty}^{-1} a_n(z - z_0)^n$ and $\sum_{0}^{\infty} a_n(z - z_0)^n$ converge?
3. If $f$ has an isolated singularity at $z = z_0$ and $\lim_{z \rightarrow z_0} f(z)$ exists, what can you conclude about the behavior of $f$ near $z_0$?
4. Does it make sense to say that an isolated singularity has a pole of order $N$ if $N$ is not an integer?
5. Does it make sense to regard a mereomorphic function as a mapping to the Riemann sphere? What about a function with an essential singularity?

# Tues. Mar 27

Read: VI - 3, 4, 5
Questions:
1. Can you express the partial fractions decomposition algebraically as giving a basis for the mereomorphic functions on $\mathbb{C}^*$?
2. To find the partial fractions decomposition of a rational function $\frac{P(z)}{Q(z)}$, do you have to know the roots of $Q$?
3. What is the relationship between the theorem on p. 182 and the theorem on p. 183? How do the hypothesis differ? How do the conclusions differ? Is one theorem a special case of the other?
4. How do the periods of a doubly periodic mereomorphic function relate to the tilings of the plane by translates of a parallelogram?

# Thurs. Mar 29

Read: VI - 6
Questions:
1. If $f(\theta)$ gives the boundary values of a function $f(z)$ analytic in the unit disc, what does this tell you about the Fourier coefficients of $f$?
2. What is the relationship between the Fourier series of the functions $f\left(e^{i \theta}\right)$ and $f\left(e^{i( \theta + s)}\right)$ ?
3. What does (6.2) tell you about the family of functions $\left( \frac{1}{\sqrt{2} \pi} e^{i k \theta} \right)$ ? How does this relate to Bessel's inequality?
4. Suppose that the Fourier coefficients of $f$ satisfy $k^n a_k \rightarrow 0$ as $k \rightarrow \pm \infty$ , for some value $n$. What does this tell you about the differentiability of $f$ ?

# Tues. Apr 10

Read: VII - 1, 2, 3
Questions:
1. What does the residue theorem say if $f$ has exactly one singularity?
2. How is Rule 4 a "special case" of Rule 3?
3. Do Rules 1-4 enable you to compute residues of any rational function? Do you need to be able to factor the denominator?
4. Could the integral $\int_{-\infty}^{\infty} \frac{1}{1+x^2} dx$ also be evaluated using a contour in the lower half-plane ? What about the integral $\int_{- \infty}^{\infty} \frac{\cos(ax)}{1+x^2} dx$?
5. In (2.5), why do we require $a>0$ ? What is the limit as $a \rightarrow \infty$ ? Can you give an intuitive explanation?
6. Could we derive (3.3) by the same method used to derive (3.2)?

# Thurs. Apr 12

Read: VII - 4, 5, 6
Questions:
1. In handling the contributions along the slit in the keyhole domain on p. 206, why does the minus sign appear?
2. For the integral in section 5, why do the integrals over $\Gamma_R$ and $\gamma_{\delta}$ vanish in the limit?
3. Could you compute the principal value integral in the example on p.213 by ordinary calculus using the partial fraction expression for $\frac{1}{x^3-1}$ ?
4. What does the Hilbert transform look like if you make the change of variables $s \rightarrow s+t$ ?

# Tues. Apr 17

Read: VIII - 1, 2, 3
Questions:
1. What is the relationship between the Theorem on p. 224 and the Theorem on p. 226?
2. If $\gamma (z)$ is a path, is $f\left( \gamma (z) \right)$ also a path?
3. How does the Argument Principle yield an algorithm for locating the zeroes of a polynomial, and hence a constructive proof of the fundamental theorem of algebra?
4. On the middle of p. 230 we choose $z^{m}$ to be the dominant term for the general monic polynomial $p(z) = z^m + a_{m-1} z^{m-1} + ... + a_0$, while on p. 229 we choose $9 z^4$ as the dominant term for the polynomial $z^6 + 9 z^4 + z^3 + 2z + 4$. What is going on here?
5. Give an informal description of Rouche's Theorem in terms of walking a dog around a lamppost with a variable length leash.
6. In what sense is Rouche's Theorem just a topological result?
7. What does the expression "converges normally" mean in Hurwitz's Theorem?

# Thurs, Apr 19

Read: VIII - 4, 5
Questions:
1. Consider the function $f(x) = x^2$ mapping $\mathbb{R}$ to $\mathbb{R}$. Does the analog of the open mapping theorem hold?
2. In what sense is the proof of the inverse function theorem constructive? What exactly is the explicit disc on which the inverse function is defined?
3. How are critical points of analytic functions related to local maxima and minima?
4. In the paragraph at the top of p. 237, what role does the function $g(z)$ play?
5. What are the angles of the "pie slices" in the diagram on p. 237?
6. How does the behavior of $f(z)$ at a critical point differ from its behavior at nearby points?

# Tues., Apr 24

Read: VIII - 6, 8
Questions:
1. How does the winding number (6.1) relate to the change of the logarithm function $log(z-z_0)$ around the curve?
2. If a closed curve $\gamma_1$ is deformed continously to a curve $\gamma_2$ in a domain $D$, can the winding number about a point not in $D$ change? Conversely, if $\gamma_1$ and $\gamma_2$ are closed curves in $D$ that have the same winding numbers about all points in $D$, must $\gamma_2$ be obtained from $\gamma_1$ by a continuous deformation?
3. In the Theorem on p.244, does the hypothesis $W\left( \gamma, \zeta \right)=0$ for all $\zeta \in \mathbb{C}\setminus D$ imply that the winding number in the formula has to be zero?
4. How does the equivalence of (i) and (iv) in the Theorem on p.254 relate to question 2 above?
5. Many theorems earlier in the book were proved under the assumption that a domain is star-shaped. Do all these theorems hold for simply connected domains?