Complex Analysis Notes

Math 444 - Fall 2024

Week 1 Notes

Tentative Schedule

Mon, Aug 26

Today we introduced the complex numbers $\mathbb{C}$ which are a field. We defined the real and imaginary parts of a complex number and also the absolute value and argument of a complex number. We did the following examples.

1. Find the real and imaginary parts of $\dfrac{1}{3+4i}$ by rationalizing the denominator.

2. Factor the polynomial $z^2 + 3iz - 2$.

3. The polynomial $z^2 + 1$ is irreducible over $\mathbb{R}$, but not over $\mathbb{C}$. Show this by factoring it over $\mathbb{C}$.

Later in the course we will use complex analysis to prove one of most important theorems in algebra:

We finished with a discussion of the polar form $z = r (\cos \theta + i \sin \theta)$ of a complex number.

Wed, Aug 28

Following Beck et. al., we will define $$e^{i \theta} := \cos \theta + i \sin \theta.$$ Later, when we define the complex exponential function, we will revisit this definition. For now, we will do some calculations that suggest the definition is a good one.

1. Use the angle addition formulas to show that $$(e^{i \alpha}) (e^{i \beta}) = e^{i (\alpha + \beta)}.$$

2. Simplify $\frac{1}{e^{i \theta}}$.

3. Calculate $\dfrac{d}{d \theta} e^{i \theta}$. Hint: Use the definition of $e^{i \theta}$ and treat $i$ like any other constant.

4. What is $(e^{i \theta})^n$?

5. What is $\sqrt{i}$?

6. Explain why $\dfrac{d}{d \theta} e^{i \theta} = i e^{i \theta}$ makes sense in the context of the velocity of a point moving counterclockwise along the unit circle.

After introducing $e^{i\theta}$, we discussed the n-th roots of unity which are the complex numbers $z$ such that $z^n = 1$. They are given by the formula $$e^{2 \pi i k/ n} \text{ where } k \in \{0, 1, \ldots, n-1\}.$$

7. Find the 3rd roots of 8.

8. Find the 3rd roots of $\left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right)$.

We finished with a discussion of square and cube roots of complex numbers and how they relate to roots of unity. We also defined the complex conjugate of $z = a+ib$ to be $\bar{z} = a - ib$.

Fri, Aug 30

We reviewed some of the useful formulas involving complex conjugates including the following:

• $|z|^2 = z \cdot \bar{z}$

• $\bar{z + w} = \bar{z} + \bar{w}$ and $\bar{ z \cdot w} = \bar{z} \cdot \bar{w}$

• $\operatorname{Re}(z) = \frac{z+ \bar{z}}{2}$ and $\operatorname{Im}(z) = \frac{z - \bar{z}}{2i}$

We also introduced and proved the triangle inequality for complex numbers $$| z + w | \le |z| + |w|.$$

1. Use the triangle inequality to prove the reverse triangle inequality $$| z - w | \ge |z| - |w|$$ by introducing a substitution $u = z - w$.

2. If $|z - w| \le \epsilon$, prove that $$\left| \frac{1}{z} - \frac{1}{w} \right| \le \frac{\epsilon}{|z| \, |w|}.$$

A convex combination of $z, w \in \mathbb{C}$ is any point $$t z + (1-t) w, ~ 0 \le t \le 1.$$ The set of all convex combinations of $z$ and $w$ is a line segment connecting $z$ to $w$, and the formula for the convex combinations is a parametric formula for the line segment. If you allow any $t \in \mathbb{R}$, then you get an affine combination of $z$ and $w$. The set of all affine combinations of $z$ and $w$ is line passing through $z$ and $w$.

Week 2 Notes

Tentative Schedule

Wed, Sep 4

Let $B_R(z) = \{w \in \mathbb{C}: |z-w| < R\}$ denote the open disk of radius $R$ around $z$. We used $B_R(z)$ to define interior and boundary points for a subset of $\mathbb{C}$. We proved the following trichotomy result

We also defined open and closed sets, and proved the following theorem.

1. If a set is both open and closed, then it can’t have any boundary points, since it includes its boundary (because it is closed) but every point is an interior point (because it is open). Are there any sets with no boundary points?

We also defined bounded sets. We finished by talking about paths which are continuous functions $\gamma: [a,b] \rightarrow \mathbb{C}$. The range of a path is a subset of $\mathbb{C}$ called a curve (note that not all books follow this terminology). We defined the derivative of a path, and discussed how to define a smooth path so that it matches our intuition for a curve with no sharp turns.

2. Why isn’t the path $\gamma(t) = t^2 + i t^3$ for $t \in [-1,1]$ smooth?

A path $\gamma: [a,b] \rightarrow \mathbb{C}$ is simple if $\gamma(t_1) \ne \gamma(t_2)$ for all $t_1 \ne t_2$, except possibly at the endpoints $t_1 = a$ and $t_2 = b$. Intuitively, a path is simple if it cannot cross itself, except possibly at the endpoints. A path is closed if $\gamma(a) = \gamma(b)$.

Fri, Sep 6

A set $A \subseteq \mathbb{C}$ is path connected if for any $w, z \in A$, there is a (continuous) path $\gamma: [a,b] \rightarrow A$ such that $\gamma(a) = w$ and $\gamma(b) = z$. The following theorem seems obvious, but it is actually famously tricky to prove (see https://en.wikipedia.org/wiki/Jordan_curve_theorem).

A sequence is a function $s: \mathbb{N}\rightarrow \mathbb{C}$. We use the notation $s_n$ to mean the same thing as $s(n)$ for sequences. A sequence $s_n$ converges to $L \in \mathbb{C}$ if for every $\epsilon > 0$, there is an $N > 0$ such that $|s_n - L | < \epsilon$ for every $n \ge N$. Intuitively, this means that for every open disk around $L$, there can only be a finite number of $n$ such that $s_n$ is not in the disk. When a sequence converges, the number it converges to is called its limit.

If we don’t know the limit of a sequence, we can still use Cauchy’s criterion to show that it must converge. A Cauchy sequence is a sequence $s_n$ such that for every $\epsilon > 0$, there exists $N > 0$ such that $|s_n - s_m | < \epsilon$ whenever $m, n \ge N$.

We talked about how the Cauchy criterion applies to sequences of real numbers and the complex numbers, in fact the property that Cauchy sequences converge in a set is known as completeness and it is one of the defining properties of the real numbers.

We used completeness to prove the following theorem about real number sequence:

We ran out of time, but a related application of completeness is the following result.

Week 3 Notes

Tentative Schedule

Mon, Sep 9

The complex numbers are a metric space because they have a distance function $d(z,w) = |z-w|$ that satisfies the following properties for all $x, y, z \in C$:

1. Definiteness $d(x,x) = 0$
2. Positivity $d(x,y) > 0$ if $x \ne y$
3. Symmetry $d(x,y) = d(y,x)$
4. Triangle Inequality $d(x,z) \le d(x,y) + d(y,z)$

It helps when working with absolute values in $\mathbb{C}$ to remember that $|z-w|$ is the distance between $z$ and $w$. We didn’t have time for a proof of the last theorem from Friday, so we worked through a proof using the fact that $\mathbb{C}$ is a complete metric space today. Recall that complete means that all Cauchy sequences converge. We proved the theorem by proving the following claims:

1. Any line segment that connects a point in $A$ to a point in $A^c$ contains a line segment that is half as long and also connects a point in $A$ to a point in $A^c$.

2. The midpoints of the line segments above form a Cauchy sequence.

3. The limit of the midpoints is a boundary point of $A$.

This proof definitely doesn’t work without assuming completeness. For example, if $A = \{x \in \mathbb{Q} : x < \pi \}$, then $A$ has no boundary points in $\mathbb{Q}$. After discussing that example, we talked about infinite series.

A series of complex numbers $\sum_{k = 0}^\infty a_k$ converges if its sequence of partial sums $S_n = \sum_{k = 0}^n a_k$ converges. It helps to know some example infinite series, so we talked about these three:

• Zeno’s series $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \ldots$.

• Grandi’s series $1 - 1 + 1 - 1 + 1 - 1 + \ldots$.

• Harmonic series $1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} + \ldots$.

Then we reviewed geometric series which are series of the form $\sum_{k = 0}^\infty a r^k$ where $a$ is the first term and $r$ is the common ratio. We proved:

Wed, Sep 11

Class was canceled since I was out with COVID.

Fri, Sep 13

• Workshop: Infinite Series

Week 4 Notes

Tentative Schedule

Mon, Sep 16

Today we reviewed the questions from the infinite series workshop. Then we talked about functions $f: \mathbb{C}\rightarrow \mathbb{C}$. We defined limits of functions several ways, and we did the following examples.

1. $\lim_{z \rightarrow 0} \frac{\bar{z}}{z}$.

2. Exercise 2.6

3. Use the $\epsilon-\delta$ definition of limits to prove that if $\lim_{z \rightarrow z_0} f(z) = L \ne 0$ then $\lim_{z \rightarrow z_0} \frac{1}{f(z)} = \frac{1}{L}$.

   Consider $$\left| \frac{1}{f(z)} - \frac{1}{L} \right| = \frac{|L - f(z)|} {|L f(z)|}$$ We can use the fact that $f(z) \rightarrow L$ to make the top as small as we want. We just need to make sure that the bottom doesn’t get close to zero at the same time. The trick is to use the triangle inequality to show that $|f(z)| > |L| - \epsilon$ when $|L - f(z)| < \epsilon$,

4. Use the sequential definition of limits to prove the product rule for limits, i.e., $\lim_{z \rightarrow z_0} f(z) g(z) = (\lim_{z \rightarrow z_0} f(z)) (\lim_{z \rightarrow z_0} g(z) )$.

Wed, Sep 18

We introduced the complex derivative for functions $f: D \rightarrow \mathbb{C}$:

$$f'(z) = \lim_{h \rightarrow 0} \frac{f(z+h) - f(z)}{h}.$$

1. Find the derivative of $f(z) = z^2$.

What does it mean for a function to be differentiable? For functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, we say that $f$ is differentiable at a vector $x \in \mathbb{R}^n$ if there is a matrix $J$ (called the Jacobian) such that $$f(x + \Delta x) \approx f(x) + J \Delta x$$ more specifically, $$\lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - (f(x) + J \Delta x)}{\|\Delta \|} = 0.$$ The expression $f(x) + J \Delta x$ is an affine linear approximation of $f$ near $x$.

3. Show that multiplication by $a + ib$ is a linear transformation on $\mathbb{C}$ that corresponds to multiplying vectors in $\mathbb{R}^2$ by the matrix $$\begin{pmatrix} a & -b \\ b & a \end{pmatrix}.$$ Note that the multiplication by a complex number can rotate and/or scale, but it cannot reflect or skew.

A complex function $f: \mathbb{C}\rightarrow \mathbb{C}$ can be thought of as a real function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, and $f'(z)$ corresponds to the Jacobian matrix $$J = \begin{pmatrix} \partial \operatorname{Re}f / \partial x & \partial \operatorname{Re}f/ \partial y \\ \partial \operatorname{Im}f / \partial x & \partial \operatorname{Im}f/ \partial y \end{pmatrix}$$

4. Observe that if $z = x+iy$, then $z^2 = x^2 - y^2 + 2 x y i$. Find the Jacobian of the function $$f(x,y) = \begin{pmatrix} x^2 - y^2 \\ 2xy \end{pmatrix}$$ and show that it corresponds to the complex number $2z$.

5. Show that the function $f(z) = \bar{z}$ is not complex differentiable. Explain why that makes sense, even though the corresponding map on $\mathbb{R}^2$, $f(x,y) = \begin{pmatrix} x \\ -y \end{pmatrix}$, does have a Jacobian matrix.

Fri, Sep 20

We introduced the Cauchy-Riemann equations.

1. $f(z) = e^z$

2. This is Exercise 2.19 in the book: Define $f(z) = 0$ if $z$ is on either purely real or purely imaginary, and $f(z) = 1$ otherwise. Show that $f$ satisfies the Cauchy–Riemann equations at $z = 0$, yet $f$ is not differentiable at $z = 0$. Why doesn’t this contradict the theorem about a complex function being complex differentiable if and only if it satisfies the Cauchy-Riemann equations?

We say that a function $f$ that is complex differentiable in an open disk around $z_0 \in \mathbb{C}$ is holomorphic at $z_0$.

Then we talked about smooth paths which are paths that are differentiable. If $\gamma$ is a smooth path in $\mathbb{C}$ and $f$ is holomorphic at every point on the curve, then the chain rule holds and says that $$\frac{d}{dt} f(\gamma(t)) = f'(\gamma(t)) \cdot \gamma'(t).$$

Idea. Notice that the angle of the tangent vector gets rotated by the argument of $f'(\gamma(t))$. If two different smooth curves both intersect at a point $z_0$, and $f$ is holomorphic at $z_0$, then since their tangent vectors are both rotated by the same angle, the angle between the two tangent vectors stays the same. This implies that holomorphic maps are conformal, that is they preserve angles between curves.

3. $f(z) = z^2 / 5$ (see Complex Grapher)

We also looked at the function $e^z$ on the complex grapher and saw that it was also conformal.

Week 5 Notes

Tentative Schedule

Mon, Sep 23

We started by reviewing the properties of the exponential function including its algebraic properties, domain, range and that it is periodic with period $2\pi i$. Then we introduced the complex natural logarithm which is the inverse of $e^z$. There is one problem with defining the inverse: $e^z$ is an $\infty$-to-1 function, every $w \in \mathbb{C}\backslash \{0\}$ has infinitely many pre-images. So we have two options:

1. $\log z$ denotes the multivalued inverse of $e^z$. It has the form $$\log z = \ln |z| + i \operatorname{arg}z$$ where $\operatorname{arg}z$ is the multivalued argument function.

2. $\operatorname{Log}z$ is the principal branch of $\log z$. It is a single valued function of the form $$\operatorname{Log}z = \ln |z| + i \operatorname{Arg}z$$ where $\operatorname{Arg}z$ is the single valued principal branch of the argument function that takes values in $(-\pi, \pi]$.

We did the following exercises.

1. Calculate $\log(1+i)$.

2. If $z = x+iy$ where $x > 0$, then $\operatorname{Arg}z = \arctan(y/x)$. Calculate the Cauchy-Riemann equations for $\operatorname{Log}z$ to verify that $\operatorname{Log}z$ is complex differentiable when $x > 0$ (in fact it is complex differentiable everywhere except at its branch cut $(-\infty, 0]$). What is the derivative of $\operatorname{Log}z$?

Wed, Sep 25

We looked at the Cauchy-Riemann equations for $\operatorname{Log}z = \tfrac{1}{2} \ln(x^2 + y^2) + i \arctan \left( \tfrac{y}{x} \right)$ again. We also did the following.

1. Give an example where $\operatorname{Log}(z+w) \ne \operatorname{Log}z + \operatorname{Log}w$.

We introduced principle values of a complex power $z^\alpha = e^{\alpha \operatorname{Log}z}$ for any $\alpha, z \in \mathbb{C}$, with $z \ne 0$. We calculated the following examples.

2. $i^{1/3}$

3. $(1+i)^{2-i}$

4. Find all other values of $(1+i)^{2-i}$ in addition to the principal value.

5. Find all solutions of the equation $e^{1/z} = i$.

We finished by talking about the reciprocal map $f(z) = \tfrac{1}{z}$. We looked at how it appears to transform lines & circles into lines and circles. We’ll look at why next time.

Fri, Sep 27

We started by proving this theorem about the reciprocal map.

The key to the proof is the fact that the solution of the algebraic equation (with real coefficients) $$a(x^2 + y^2) + b_1 x + b_2 y + c = 0$$ is a circle (possibly degenerate to a point or $\varnothing$) if $a \ne 0$, and it is a line if $a = 0$.

It helps to think of the reciprocal map $f(z) = \tfrac{1}{z}$ as a bijection (1-to-1 and onto map) from $\mathbb{C}\cup \{ \infty\} \rightarrow \mathbb{C}\cup \{ \infty \}$. We define $$\frac{1}{\infty} = 0 \text{ and } \frac{1}{0} = \infty.$$ We call $\mathbb{C}\cup \{ \infty \}$ the extended complex plane.

The reciprocal map is a special case of an important family of bijections on $\mathbb{C}\cup \{\infty\}$ called Möbius transforms. A Möbius transform (also known as a Linear Fractional Transform) is a map $$f(z) = \frac{a z + b}{c z + d}$$ where $a, b, c, d \in \mathbb{C}$ satisfy $ad - bc \ne 0$. We proved the following facts.

1. A Möbius transform always has two fixed points in $\mathbb{C}\cup \{ \infty \}$.

2. For any invertible matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$, we can define an associated Möbius transform $$T_A(z) = \frac{a_{11} z + a_{12}}{a_{21} z + a_{22}}.$$ Then if $A, B \in \mathbb{C}^{2 \times 2}$ are any two intertible matrices, $$T_A \circ T_B = T_{AB}.$$

In particular the inverse of a Möbius transform can be found by inverting its matrix: $$T_A^{-1} = T_{A^{-1}}.$$ Notice also that if you multiply a matrix by a constant, the Möbius transform doesn’t change, so $$T_{cA} = T_A.$$ That is convenient because the inverse of a 2-by-2 matrix is $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$$ Therefore you can invert a Möbius transform $$f(z) = \frac{a z + b}{c z + d}$$ by swapping $a$ and $d$ and making $b$ and $c$ negative: $$f^{-1}(z) = \frac{d z - b}{-cz + a}.$$

Week 6 Notes

Tentative Schedule

We started by proving that

Theorem. Any Möbius transformation maps lines & circles to lines & circles.

To prove this, we showed that if $c \ne 0$, then $$T(z) = \frac{az + b}{cz + d}$$ can be expressed as a composition of three maps $$T_3(z) = \left(\frac{bc-da}{c} \right) z + \frac{a}{c}, ~~~~~ T_2(z) = \frac{1}{z}, \text{ and } T_1(z) = cz + d.$$

Theorem 2. A Möbius transform is uniquely determined by where it maps any three points in $\mathbb{C}\cup \{ \infty \}.$

Proof. If $z_1, z_2, z_3$ are any three distinct points in $\mathbb{C}\cup \{\infty\}$, suppose that there are two Möbius transforms $T$ and $S$ such that $S(z_i) = T(z_i)$ for each $i$. Then $S^{-1} T$ has three distinct fixed points which is impossible unless $S^{-1} = T^{-1}$ which means that $S$ and $T$ are the same. $\square$

1. Find a Möbius transform that sends $\infty$ to 2, $i$ to $\infty$, and $1$ to $1$.

2. Find a general formula for a Möbius transform that sends $z_1, z_2, z_3$ to $w_1$, $w_2$, $w_3$. Hint: It might help to start with a formula to send $z_1$ to $0$, $z_2$ to $1$ and $z_3$ to $\infty$.

3. Find a Möbius transform that leaves $+1$ and $-1$ fixed but maps 0 to $\infty$. Actually, this wasn’t a great example because it turned out to be the reciprocal map!

4. How does the Möbius transform from the previous example transform the circles in this figure where the blue circle is centered at $i$?

Video: Möbius Transformations Revealed

Wed, Oct 2

We reviewed for the exam by talking about these two problems.

1. How does the Möbius transform $T(z) = \dfrac{z-i}{z-1}$ transform the three shapes shown below?

2. Suppose $s_n$ is a sequence in $\mathbb{C}$ such that $|s_{n+1} - s_{m + 1}| \le 0.9 |s_n - s_m|$ for all $m, n \in \mathbb{N}$.

   1. Show that $s_n$ is bounded by finding an upper bound for $|s_n|$.

   2. Show that $s_n$ is a Cauchy sequence by finding an upper bound for $|s_n - s_m|$ when $n > m$.

Week 7 Notes

Tentative Schedule

Mon, Oct 7

We introduced complex integrals, which are defined for any piecewise smooth path $\gamma: [a, b] \rightarrow \mathbb{C}$ by $$\int_\gamma f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt.$$ We talked about why this definition makes sense and we did these examples.

1. $\int_\gamma z^2 \, dz$ on the upper half of the unit circle from $1$ to $-1$. We started by using Python to numerically approximate the integral with a Riemann sum:

from cmath import *

n = 1000
total = 0
for k in range(1000):
    delta_z = exp(1j * pi * (k + 1) / n) - exp(1j * pi * k / n)
    z = exp(1j * pi * k / n)
    total += z ** 2 * delta_z
    
print(total)

2. $\int_\gamma (\overline{z})^2 \, dz$ on the path $\gamma(t) = t(1+i)$ with $t \in [0,1]$.

3. $\int_\delta (\overline{z})^2 \, dz$ on the path $\delta(t) = t+it^2$ with $t \in [0,1]$. (Link: Sympy code)

from sympy import *
x, y, z = symbols('x y z')
t = symbols('t',real=True)

f = conjugate(z)**2
g = t+t**2*I

print(integrate(f.subs(z,g)*diff(g,t),(t,0,1)))

4. Use the formula $\operatorname{length}(\gamma) = \int_a^b |\gamma'(t)| \, dt$ to find the length of the unit circle.

Wed, Oct 7

Today we introduced

We defined what simply connected means (intuitively it means that $D$ has no holes), and we reviewed simple closed curves. We also talked about why the specific parametrization of a curve does not matter for integrals (see Proposition 4.2), but the orientation does, and the symbol $\oint$ indicates that the orientation of a simple closed curve is positive (counterclockwise). We calculated

1. $\oint_{|z| = 1} \frac{1}{z} \, dz$

and saw that Cauchy’s theorem does not apply because the reciprocal function is not holomorphic at 0.

Then we talked about how to prove Cauchy’s theorem using

Here is a nice video explanation of the intuition behind Green’s theorem. One problem with using Green’s theorem to prove Cauchy’s theorem is that it requires us to assume that $f'(z)$ is continuous (so that the partial derivatives are all continuous. It turns out that is true for all holomorphic functions, as we will see later. The Complex Variables book has a different proof of Cauchy’s theorem that doesn’t require this assumption.

We finished by starting to talk about how useful Cauchy’s theorem is. One application of Cauchy’s theorem is that it implies that every holomorphic function has a holomorphic antiderivative. You can define the antidervative $F(z)$ of a holomorphic function $f$ on a simply connected open domain this way: $$F(z) = \int_\gamma f(z) \, dz$$ where $\gamma$ is any piecewise smooth path from a fixed $z_0 \in D$ to $z$. We talked about why $F(z)$ is well-defined.

Friday, Oct 11

Last time we showed that when a function $f(z)$ is holomorphic on a simply connected open set $D$, then the integral on any two paths $\gamma_1$ and $\gamma_2$ that both start at a point $z_0$ and end at $z_1$ and stay inside $D$ must be the same:

$$\int_{\gamma_1} f(z) \, dz = \int_{\gamma_2} f(z) \, dz.$$

In other words, the value of the integral is independent of the path. Today we proved some important theorems to make working with complex integeral easier.

This theorem follows immediately from the definition of a complex path integral and the chain rule, combined with the evaluation theorem from real variable calculus. An application of this theorem is the following.

1. Calculate $\int_\gamma z^n \, dz$ for any piecewise smooth path $\gamma$ that begins at $z \in \mathbb{C}$ and ends at $w \in \mathbb{C}$. Note: this solution works as long as $n \ne -1$ with the caveat that the path shouldn’t pass through 0 if $n$ is negative. Why won’t this work if $n = -1$?

This is like a triangle inequality for complex integrals. In fact, you can use the triangle inequality to prove it.

Last time we used independence of path to define $$F(z) = \int_\gamma f(z) \, dz$$ where $\gamma$ is any path in $D$ from $z_0$ to $z$. We just have to show that $F'(w) = f(w)$ for every $w \in D$ to prove that this function $F$ is the antiderivative needed for the antiderivative theorem. Two key ideas in the proof are the following:

1. Let $\delta$ be a parametrization of the line segment from $w$ to $w+h$ in $D$. Show that $\int_\delta f(w) \, dz = h f(w)$.

2. Use the ML-inequality to estimate $$\left| \frac{\int_\delta f(z) - f(w) \, dz}{h} \right|.$$

Week 8 Notes

Tentative Schedule

Wed, Oct 16

That’s kind of weird if you think about it. This theorem says that the value of $f$ at a point inside a curve is completely determined by the values of $f$ on the curve. The proof required two key insights:

1. If $f$ is complex differentiable at $w$, then $$f(z) = f(w) + f'(w) (z-w) + \epsilon(z),$$ where the error term $\epsilon(z)$ satisfies $\lim_{z \rightarrow w} \frac{\epsilon(z)}{z-w} = 0$.

2. If $C_1$ and $C_2$ are two different simple, closed, piecewise smooth curves in a region where $f$ is holomorphic, then $$\oint_{C_1} \frac{f(z)}{z-w} \, dz = \oint_{C_2} \frac{f(z)}{z-w} \, dz.$$

Use the Cauchy integral formula to evaluate the following integrals.

1. $\oint_C \frac{e^z}{z-1} \, dz$ where $C$ is the square with vertices at $10, 10i, -10, -10i$. (https://youtu.be/NJap6Vm5mEk)

2. $\oint_{|z-i|=1} \frac{1}{z^2+1} \, dz$

3. $\oint_{|z|=3} \frac{e^z}{z^2 - 2z} \, dz$

4. $\oint_{|z| = 1} \frac{z^2+1}{z(2z+1)} \, dz$. (https://youtu.be/APoh2B5S2ok)

We finished by looking at the following application of the Cauchy integral formula:

5. $\int_{-\infty}^\infty \frac{1}{x^4 + 1} \, dx$.

The key is to integrate over the boundary of the upper half of the disk of radius $R$ centered at the origin (where $R$ is very large). Then use the ML-inequality to show that the integral over the circular part of the boundary goes to zero, leaving just the real integral we want.

Friday, Oct 18

Last time we ran into the problem of figuring out how to decompose functions like

$$g(z) = \frac{1}{z^4 + 1} \text{ into the form } \frac{f(z)}{z - w}.$$

That isn’t always easy to do. Here is a shortcut that you can use. If $g(z) = \frac{f(z)}{z-w}$ where $f$ is holomorphic in a neighborhood around $w$, then $f(z) = g(z) \cdot (z-w)$ for every $z \ne w$, and since holomorphic functions are continuous: $$f(w) = \lim_{z \rightarrow w} g(z) \cdot (z - w).$$ The value of this limit is called the residue of $g(z)$ at $w$. You can often use L’Hospital’s rule to find the residue.

1. Find the residues of $\dfrac{1}{z^4+1}$ at $w_1 = e^{\pi i/ 4}$ and $w_2 = e^{3 \pi i /4}$.

A function $f:D \rightarrow \mathbb{C}$ is analytic on an open domain $D \subseteq \mathbb{C}$ if it is infinitely (complex) differentiable and for every $w \in D$, there is a power series $\sum_{k = 0}^\infty a_k (z-w)^k$ that absolutely converges to $f(z)$ for every $z$ in an open disk around $w$.

And immediate corollary is:

The key to proving the first theorem is to confirm that the following expression converges when $|u - w| < R$. $$2\pi i f(u) = \oint \frac{f(z)}{z - u} \, dz = \oint \frac{f(z)}{(z - w) - (u - w)} \, dz = \oint \frac{\frac{f(z)}{(z-w)}}{1 - \left( \frac{u - w}{z-w}\right)} \, dz$$ $$= \oint \sum_{k = 0}^\infty \frac{f(z)}{(z-w)^{k+1}} (u-w)^k \, dz =\sum_{k = 0}^\infty \oint \frac{f(z)}{(z-w)^{k+1}} (u-w)^k \, dz.$$

We ran out of time before we could do this exercise, but we’ll start with it on Monday.

2. $\oint_{|z| = 3} \frac{e^{iz}}{(z+i)^2} \, dz$. (https://youtu.be/WJOf4PfoHow)

Week 9 Notes

Tentative Schedule

Mon, Oct 21

We ran out of time last class before we could do this example.

1. $\oint_{|z| = 3} \frac{e^{iz}}{(z+i)^2} \, dz$. (https://youtu.be/WJOf4PfoHow)

Kade asked what happens if the function $f(z)$ in Cauchy’s integral formula is not holomorphic everywhere in the domain, which was a great questions, so we looked at a variant of the previous problem where we replaced the function $e^{iz}$ with the function $1/z$.

2. $\oint_{|z| = 3} \frac{1/z}{(z+i)^2} \, dz$.

Then we talked about the following partial converse to Cauchy’s theorem.

We outlined the proof by reviewing the proof of the antiderivative theorem. Since integrals of $f$ are independent of path in $D$, we can define an antiderivative $F(w) = \int_{z_0}^w f(z) \, dz.$ The same proof as before shows that the antiderivative is holomorphic, but Cauchy’s integral formula for derivatives implies that $f$ is also complex differentiable in $D$, so it is also holomorphic.

After that, we reviewed several important facts about power series.

Another fact which we stated but did not prove is

You can use Morera’s theorem to prove this. But the proof also requires knowing that analytic functions are continuous in an open disk too. The easiest way to prove that is to argue that the partial sums of the Taylor series converge uniformly to $f(z)$ in the disk, but we haven’t talked about how continuity is preserved under uniform convergence.

We left the following exercise to try on your own.

4. Suppose that the power series $\sum_{k = 0}^\infty a_k z^k$ has radius of convergence $R$. Prove that its derivative $\sum_{k = 0}^\infty k a_k z^{k-1}$ also has radius of convergence $R$.

Wed, Oct 23

The winding number of a (piecewise smooth) closed path $\gamma$ around a point $w \in \mathbb{C}$ is $$\frac{1}{2\pi i} \int_\gamma \frac{1}{z-w} \, dz$$ (assuming that $\gamma$ does not intersect $w$).

We used the (now updated) complex grapher to look at some examples. Then we proved this theorem by doing the following exercises in class:

1. If $\gamma(t) = p(e^{it})$, show that $$\int_\gamma \frac{1}{z} \, z = \oint_{|z|=1} \frac{p'(z)}{p(z)} \, dz.$$

2. For a polynomial $$p(z) = (z-r_1)^{m_1} (z-r_2)^{m_2} \cdots (z-r_k)^{m_k},$$ show that $$\frac{p'(z)}{p(z)} = \frac{m_1}{z-r_1} + \frac{m_2}{z-r_2} + \ldots + \frac{m_k}{z-r_k}.$$ Hint: Use mathematical induction on $k$ if you want to make this rigorous.

We can use this theorem to derive one proof of the Fundamental Theorem of Algebra.

Assume without loss of generality that $p$ is a monic (leading coefficient is one) degree $n$ polynomial. Note that $p$ has a root if and only if the polynomial $p(Rz)/R^n$ has a root for any $R > 0$.

3. Show that there exists $R$ large enough so that $|z^n - \frac{p(Rz)}{R^n}| < 1$ for all $z$ on the unit circle.

You can get a proof of the Fundamental Theorem of Algebra by combining this inequality with the following intuitive result from topology.

Fri, Oct 25

We started by talking about Homework 9. We did the following example, which is similar to problem #4:

1. Find the Maclaurin series for the function $$\frac{1}{z^3 - 8}.$$

A function $f: \mathbb{C}\rightarrow \mathbb{C}$ is entire if it is holomorphic everywhere in $\mathbb{C}$.

To prove Liouville’s theorem, we need a quick observation and an idea. The observation is that if $f'(z) = 0$ everywhere, then $f$ must be constant. The idea is that we can use Cauchy’s integral formula for derivatives to calculate $f'(z)$ if $f$ is bounded and entire: $$f'(z) = \frac{1}{2\pi i} \oint_{|\xi - z| = R} \frac{f(\xi)}{(\xi - z)^2} \, d\xi.$$ What happens if the radius $R$ gets really really big? Use the max-times-length inequality to estimate $|f'(z)|$.

We finished by applying Liouville’s theorem to re-prove the Fundamental Theorem of Algebra. Unlike the proof we gave last time, using Liouville’s theorem to prove the FToA only guarantees one root. But that is enough if you use this result from algebra.

A corollary of the division algorithm and the Fundamental Theorem of Algebra is:

We used these ideas to completely factor the polynomial:

2. $p(z) = z^3 - 3z^2 + z - 3$.

Another good example to factor completely would be:

2. $p(z) = z^3 - 8$.

Week 10 Notes

Tentative Schedule

Mon, Oct 28

1. Consider the following power series for $$\operatorname{Log}(z) = (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \ldots.$$
   1. Re-write this series in $\Sigma$-notation.
   2. What are the coefficients $a_n$ of this power series?
   3. What is the center and radius of convergence?
      
   4. How would you find a Taylor series for $\operatorname{Log}(z)$ centered at $w = -1 + i$? What is the radius of convergence for that Taylor series?
2. Find power series for the following functions by using the Maclaurin series for $e^z$, $\sin z$, and $\cos z$:
   1. $e^{-z^2/2}$
   2. $\dfrac{\sin z}{z}$

Why do power series have a radius of convergence?

We proved this in class, and then we talked about why the least upper bound of the set

$$\left\{ r \ge 0 ~:~ \sum_{n = 0}^\infty a_n r^n \text{ converges} \right\}$$

must be the radius of convergence.

Another result that you can derive from the above is:

We talked about the intuition of why this is true: because there is an $N$ after which, our power series will be sandwiched between two geometric series, one with radius of convergence $R+\epsilon$ and the other with radius of convergence $R-\epsilon$.

Here are some exercises we didn’t have time for in class.

1. Use the ratio test to find the radius of convergence of the power series $$1 + z + \frac{3}{4} z^2 + \frac{4}{8} z^3 + \frac{5}{16} z^4 + \ldots.$$

2. Why can’t you use the ratio test as written to find the radius of convergence for the Maclaurin series for $\sin z$?

Wed, Oct 30

A sequence of functions $f_n: D \rightarrow \mathbb{C}$ converges uniformly to a function $f: D \rightarrow \mathbb{C}$ if for every $\epsilon > 0$, there exists $N >0$ such that $|f_n(z) - f(z)| < \epsilon$ for all $n \ge N$ and $z \in D$.

1. Show that if $f$ is a holomorphic function with a power series $\sum_{n = 0}^\infty a_n (z-w)^n$ that has radius of convergence $R > 0$, then the partial sums converge uniformly to $f$ inside any closed disk around $w$ with a radius $r < R$.

2. Show that if $f_n$ is a sequence of continuous functions that converge uniformly to $f$ on an open set $D$, then $f$ is continuous on $D$. You have to show that for any $w \in D$ and any $\epsilon > 0$, there is a $\delta > 0$ such that $$|z - w | < \delta \text{ implies } |f(z) - f(w)| < \epsilon.$$ Hint: Fix $\epsilon$. Choose $N$ big enough so that $|f_n - f| < \tfrac{\epsilon}{3}$ for all $n \ge N$. Then use the continuity of $f_n$ and the triangle inequality (see the image below).

After that we talked about the idea that if a function is defined by a power series in a open disk, then you can often analytically continue the function by constructing a new power centered at a different point inside the open disk. In this way you can extend the function past its original disk of convergence. This has one immediate consequence. If a holomorphic function $f:D \rightarrow \mathbb{C}$ on an open connected domain $D$ has $f^{n}(w) = 0$ for all $n \ge 0$ at one point $w \in D$, then $f = 0$ everywhere in $D$.

Find the orders for the following zeros:

3. $w = 0$ for $f(z) = z-\sin z$.

4. $w = \pi i$ for $g(z) = e^z + 1$.

Fri, Nov 1