Table of Contents
Holomorphic Function, Meromorphic Function
tl;dr
A function $f: \C \to \C \cup \{\infty\}$ is holomorphic at $z_0$ if there exists $\epsilon>0$, and complex numbers $a_0,a_1,\cdots$ that for all $|z - z_0|<\epsilon$, we can write $$ f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n. $$
A function $f: \C \to \C \cup \{\infty\}$ is meromorphic at $z_0$ with order $m \geq 1$ pole, if there exists $\epsilon>0$, and complex numbers $a_{-m}, \cdots , a_0,a_1,\cdots$ that for all $0 < |z - z_0|<\epsilon$, we can write $$ f(z) = \sum_{n=-m}^\infty a_n (z-z_0)^n. $$
Real function Differentiability
First, let's us recall some notion in real analysis. Given a function $f: \R \to \R$, we can talk about its derivative at a point $x_0 \in \R$. It is defined as $$ f'(x_0) = \lim_{\epsilon \to 0} \frac{f(x_0 + \epsilon) - f(x_0)} {\epsilon}. $$ If the derivative $f'(x_0)$ exists, then we say the function $f$ is differentiable at $x_0$. For example, $f(x) =|x|$ is not differentiable at $x=0$. If the derivative $f'(x_0)$ exists for all $x_0 \in \R$, then we say $f$ is differentiable on $\R$.
Ex 1 $$ f(x) = \begin{cases} x & x \geq 0 \cr 2 x & x \leq 0 \end{cases} $$ Is it differentiable at $x=0$? Can you plot $f'(x)$?
Ex 2 $$ f(x) = \begin{cases} x^2 & x \geq 0 \cr 2 x^2 & x \leq 0 \end{cases} $$
Is it differentiable at $x=0$? Can you plot $f'(x)$? (this is an example, where the function $f(x)$ is differentiable, but the derivative $f'(x)$ is not continuous, hence we cannot define $f^{“}(x)$ at $x=0$, hence $f(x)$ is not a smooth function on $\R$).
Complex function Differentiability
On the face value, the definition for complex differentiability is just replacing the word 'real' by 'complex', in the above section.
Definition: we say $f: \C \to \C$ is differentiable at $z_0 \in \C$, if $$ f'(z_0) = \lim_{\epsilon \to 0} \frac{f(z_0 + \epsilon) - f(z_0)} {\epsilon} \quad \text{ exists.} $$
Remark: Here $\epsilon$ can go to $0$ in different directions, we can let $\epsilon = r e^{i\theta}$ for fixed $\theta$ and positive $r \to 0$. The condition that derivative exists really means no matter how $\epsilon$ approach $0$, the above limit exists.
Let's see some example.
- $ f(z) = z^3 $ at $z_0$, what is $f'(z_0)$?
- $ f(z) = |z|^2 $ at $z_0 = 0$, at $|z_0|=1$. Does $f'(z_0)$ exist?
Definition: we say a subset $\Omega \In \C$ is an 'open subset of $\C$', or simply 'open', if for any point $p \in \Omega$, we can enlarge it to a small ball $B_\epsilon(p) \In \Omega$, where $B_\epsilon(p) = \{ z \mid |z-p| < \epsilon \}$.
Example: the strip $| Re(z) | < 1$ is open; the line $Re z = 1$ is not open.
Definition: We say a function $f: \Omega \to \C$ is an analytic function (aka holomorphic) on $\Omega$ if for any $p \in \Omega$, $f'(p)$ exists.
Example:
- $f(z) = 1/z$ is holomorphic on $\Omega = \C \RM \{0\}$.
- $f(z) = 1/ (z^2+1)$ is holomorphic on $\Omega = \{z \in \C \mid z^2 + 1 \neq 0 \} = \C \RM \{ \pm i \}$.
Taylor expansion
Suppose $f: \Omega \to \C$ is a holomorphic function on $\Omega$.
Then the derivative $f'(p)$ exists for every point $p$ in $\Omega$ by definition. Furthermore, $f': \Omega \to \C$ itself is a holomorphic function (a non-trivial result, not true for real differentiable function). Hence, we can differentiate $f$ anytimes we want on $\Omega$.
The Taylor expansion of $f$ centered at point $p \in \Omega$ is the following identity. If $B_r(p) \In \Omega$, then for any $z \in B_r(p)$, we have $$ f(z) = f(p) + f'(p) (z-p) + f”(p) \frac{ (z-p)^2}{2!} + \cdots + f^{(n)}(p) \frac{ (z-p)^2}{n!} + \cdots. $$
Point Singularity
Suppose $f: \C \RM \{0\} \to \C$ is holomorphic function, you cannot help but wonder, what goes wrong at $0$? You might find
- a removable singularity (i.e. a false alarm, $f$ is all good at $z=0$)
- a pole, $z^n f(z)$ is holomorphic
- an essential singularity: anything else, like $e^{1/z}$, $e^{1/z^2}$.
If you had a pole, we can apply that Taylor expansion formula (at $z=0$) to $z^n f(z)$, then divide out by $z$.
Exercises
Find the first two terms in these expansions.
1. Taylor expand $(z+1)(z+2)$ around $z=3$.
2. Laurent expand $1/[(z-1)(z-2)]$ around $z=1$. And do it again, this time around $z=2$.
2.5 (Optional) Laurent expand $e^{1/z + z}$ around $z=0$.
3. You may have heard about Cauchy Riemann equation: let $f(z)$ be a $\C$-valued function on $\C$, and let $z = x+iy$, $f = u+iv$, then we can view $u,v$ as real valued functions depending on $x,y$.
If $f$ is holomorphic, then we have $$ \frac{\d u(x,y)}{\d x} = \frac{\d v(x,y)}{\d y}, \quad \frac{\d v(x,y)}{\d x} = - \frac{\d u(x,y)}{\d y}$$
Your task: either prove this in general if you feel strong, or verify that this is true for your favorite holomorphic function (don't choose $f$ to be a constant, too boring)
Homework 3
(Due Wednesday, Sep 13)
0. Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing).
1. let $z = 2 e^{i \pi / 3}$,
- compute $z^2, z^3$.
- what is $\log z$? (be aware this is a multivalued function)
2. how many complex solution does $z^4 = -1$ have? what are they?
3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? How about $z^{1/2}$?
4. express $\sin(1+2 i)$ in terms of exponential. Is it true that $\sin(z) = Im( e^{i z})$ for all real $z$, for all complex $z$? (corrected, previous question was asking $\sin(z) = Re( e^{i z})$, which is false even for $z$ real)
5. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$?
September 8th: basic complex functions
Today we are going to meet with some old friends, which will serve as anchor when we go out and meet with more exotic ones.
1. exponential and log
2. sin, cos, sinh, cosh (not a big deal, they are linear combination of exp,log)
3. power and roots. $\sqrt{z}$? (multivalued function)
4. Taylor series, Laurent series.
Exercise
1. let $z = 2 e^{i \pi / 3}$,
- compute $z^2, z^3$.
- what is $\log z$? (be aware this is a multivalued function)
2. how many complex solution does $z^4 = -1$ have? what are they?
3. let $z = 2 e^{i \pi / 3}$. What does $z^i$ mean? is it multivalued? How about $z^{1/2}$?
4. express $\sin(1+2 i)$ in terms of exponential. Is it true that $\sin(z) = Re( e^{i z})$ for all real $z$, for all complex $z$?
5. What is the Laurent expansion (first 3 terms) of $\frac{\cos(z)}{z}$ around $z=0$? $\frac{\cos(z)}{\sin(z)}$ around $z=0$?
September 6: Introduction to complex numbers
1. what is complex number? (a pair of real numbers, put together, that you can multiply together)
2. why we need it? (I will leave that to you)
3. how to think about it? (Cartesian coordinate, polar coordinate. )
4. what can you do with it? (multiplication, addition.)
5. real part, imaginary part, modulus (absolute value), argument (aka phase angle)
6. complex conjugate. (hey, what does $\sqrt{-1}$ mean? $i$ or $-i$? does it matter? do we have a preference?)
(optional) other weird “numbers”? (if you walk like a number, talk like a number, operate like a number, I will call you a number!) The notion of a 'field'.
post lecture note
Roughly speaking, a 'ring' is a set whose elements can do addition and multiplication (among) themselves. Example: $\Z$, polynomial. (to be precise, we talk about commutative multiplication, that satisfies $x \cdot y=y\cdot x$. matrix multiplication may not be commutative.)
A 'field', is a ring where any nonzero element has a multiplicative inverse. $\Z$ is not a field. $\Q$, $\R$, $\C$ are field.
$\gdef\F{\mathbb F}$
We talked about finite field. Given a prime number $p$, we define $\F_p = \Z / p\Z$, This notation may reminds you of the quotient vector space $V/W$, indeed, $\Z / p \Z$ is the set of equivalence class, where we say two integers $n_1, n_2$ are equivalent (and write $n_1 \equiv n_2 (mod p)$), if $n_1 - n_2 \in p \Z$, i.e. the difference is a multiple of $p$. In class, we set $p=7$, and we say $1 \equiv 8 (mod 7)$. If we use $[n] = n + p \Z$ the equivalence class that $n$ belongs to, then we write $[1]=[8]$. $$\F_7 = \{ [0], [1], \cdots, [6]\}$$ We have arithematics like $$ [a] + [b] = [a+b], \quad [a] \cdot [b] = [ab]. $$ For example, $[2] \cdot [4] = [8] = [1]$. (When there is no danger of confusion, we just write $n$ for $[n]$)
Question (optional):
1. Can you write down how $\F_5$ behave? For example, what is $[2] + [4] = ?$ What is who multiply $[3]$ equasl $[1]$?
2. 'finite field version of complex number'. Take $K$ be a field. We consider $K[\sqrt{-1}] := K[x] / (x^2+1) = \{a + b x \mid a, b \in K, x^2 = -1\}$. Is this always a field? Namely, can you always define $1/(a+bx)$? We see that for $K = \R$, this works, $1/(a+bx) = (a-bx) / (a^2 + b^2)$. Does the inverse always exist for $K[\sqrt{-1}]$? Try $K = \Q, \F_5, \F_7$, see if you can find some pattern.
3. In class, we also talked about, can you define 'super complex number', that instead of using two real numbers $a,b$ to represent a complex number $a + b i$, but three real numbers? For example, we can try $$ \R[x]/(x^3-1) = \{ a+b x + c x^2 \mid a,b,c \in \R, x^3=1 \} $$ can you define multiplication on it? $ ( 1+ x + x^2) (2 + x) = ?$
Does every nonzero element has a (multiplicative) inverse? For example, $x$ has inverse, $$ 1/x = x^2. $$ $x+1$ has inverse, we have $$ \frac{1}{1+x} = \frac{1-x+x^2}{(1+x)(1-x+x^2)} = \frac{1-x+x^2}{1+x^3} = \frac{1-x+x^2}{2}. $$ Does $x-1$ has inverse?
you may ask: why we care about other 'field'? I am happy with $\C$ and $\R$. I don't have a good answer for that, maybe you will find some application some day.
Exercise
(part of homework) Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing).
Homework 1
For all the following exercises, if you feel it is too easy, skip it; if you find it interesting and relevant, do it; if you find it too hard, ask about it on discord, let's tackle it together.
1. If you are not familiar with set theory notation and terminology, watch these two short videos: 1. set notations , 2. how to describe a set (5 min each). If you need to know what is injective / surjective of map of sets, try wikipedia.
- Explain in your own word, what is a set, what is a map. Give examples.
- Can you put 'unrelated things' in a set? Like, a set that consist of three elements: $\{\text{your instructor Peng}, \text{a Gorilla}, \text{the Sun}\}$?
- When you define a map $f: A \to B$, where $A,B$ are two sets, can one element in $A$ be sent to two or more elements in $B$?
2. About linear map. The following functions $f: \R \to \R$, which is a linear map, and which is not?
- $f(x) = |x|$
- $f(x) = (x)^2$
- $f(x) = 3x$
- $f(x) = x+1$.
3. About linear subspace. Let $V = \R^2$, is the following subset $V' \In V$ a subspace? Explain why. You need to check if elements in $V'$ are closed under the vector addition and scalar multiplication. (if you are unfamiliar with the set notation, watch the two videos above.)
- $V' = \{ (x,y) \in \R^2 \mid x+y=0 \}$
- $V' = \{ (x,y) \in \R^2 \mid x+y=1 \}$
- $V' = \{ (x,y) \in \R^2 \mid x>0, y>0 \}$.
4. Matrix manipulations. Write out a $3 \times 2$ matrix $A$, and a $2 \times 2$ matrix $B$, and multiply them together $AB$. Does $BA$ make sense? What do you get?
5. Equivalence relation and equivalence classes. read about it on wiki. Explain it in your own words and give examples. https://en.wikipedia.org/wiki/Equivalence_class
6. Ask someone in your class a question. (the more question the better) Write down the question that you asked and who you asked (could be as simple as “what is $\forall$?” I hope to use this as an excuse to open up talking). You can DM people on discord.
Homework is due on Monday in class. Please write or print out your answer. The homework is graded by completion. It is good opportunity to make mistakes. If you don't do it (not because of it's easy but because you are lazy), then you don't get points.
