@Chen Siyi

October 19, 2020

Note4 Complex Analysis

Points in the Complex Plane

• For a given $z ∈ \mathbb{C}$ and $ε > 0$, the set

  $B_ε(z)=${$w ∈ \mathbb{C}$ | $|w − z| < ε$},

  is called an $ε- \text{neighborhood}$ of $z$;

  $B_ε(z)=${$w ∈ \mathbb{C}$ | $0 < |w − z| < ε$},

  is called an $ε- \text{deleted neighborhood}$ of $z$.

• A point $z_0$ is an interior point of set $S ⊂ \mathbb{C}$ if there is some $ε$ neighborhood of $z_0$ which is a subset of $S$.

• A point $z_0$ is an exterior point of a set $S ⊂ \mathbb{C}$ if there is some $ε$ neighborhood of $z_0$ containing no points of $S$ (i.e., disjoint from $S$).

• A point $z_0$ is a boundary point of set $S ⊂ \mathbb{C}$ if it is neither an interior point nor an exterior point of $S$.

• A point $z_0$ is an accumulation point of set S ⊂ C if each deleted neighborhood of $z_0$ contains at least one point of $S$.

Question1

Find the set of interior points, boundary points, accumulation points, and isolated points for:

Answer1

Sets of Points in the Complex Plane

• A set $Ω⊂\mathbb{C}$ is called open if for every $z∈Ω$ there exists an $ε>0$ such that $B_ε(z)=${$w ∈ \mathbb{C}$ | $|w − z| < ε$} $⊂ Ω$. A set is called closed if its complement is open.

• A set $Ω⊂\mathbb{C}$ is called bounded if $Ω⊂B_R(0)$ for some $R>0$.

• A set $K ⊂ \mathbb{C}$ is called compact if every sequence in $K$ has a subsequence that converges in $K$ . A set $K ⊂ \mathbb{C}$ is compact if and only if it is closed and bounded.

• An open (closed) set $Ω ⊂ \mathbb{C}$ is called disconnected if there exist two open (closed) sets $Ω_1$, $Ω_2 ⊂ \mathbb{C}$ such that $Ω_1 ∩ Ω_2 = ∅$ and $Ω = Ω_1 ∪ Ω_2$.

• If $Ω$ is not disconnected, $Ω$ is called connected. A set $Ω ⊂ \mathbb{C}$ is connected if and only if for any two points in $Ω$ there exists a curve joining them.

• An open and connected set is called a domain, or region.

• Define the diameter of a set $Ω ⊂ \mathbb{C}$ by

  $$\operatorname{diam}(\Omega):=\sup _{z, w \in \Omega}|z-w|$$

Question2

Give a set which is open and closed.

Give a set which is closed and unbounded.

Answer2

Question3

Is the set $U = \{ Z\in \mathbb{C}:2<|z|\leq 3\}$ open or closed?

Answer3

Functions in the Complex Plane

• Differentiable / holomorphic
• Analytic

Complex Differentiability

Definition of Holomorphic

We say that a function $f : \mathbb{C} → \mathbb{C}$ is complex differentiable, or holomorphic, at $z ∈ \mathbb{C}$ if

A function is holomorphic on an open set $Ω ⊂ \mathbb{C}$ if it is holomorphic at every $z ∈ Ω$. A function that is holomorphic on $\mathbb{C}$ is called entire.

Decide Holomorphic

The Cauchy-Riemann Differential Equations

If $f$ is holomorphic, then

And suppose that the partial derivatives of u and v exist, are continuous and satisfy the Cauchy-Riemann equations. Then f is holomorphic.

A Second look

Define two operators:

If $f$ is holomorphic, then

Question4

Decide whether the complex variable function f is differentiable:

Answer4

Hint: In addition to the obvious way, can you prove by the substitutions $x=\frac{z+\bar{z}}{2} \quad$ and $\quad y=\frac{z-\bar{z}}{2 i}$ ?

A Third Look

Define $u(x, y)$ to be a harmonic function if:

Define $u(x, y)$ and $v(x, y)$ to be a harmonic conjugate if:

is differentiable.

If $f$ is holomorphic, then $u, v$ are harmonic.

A Special Case-Power Series

The power series

defines a holomorphic function in its disc of convergence. The (complex) derivative of f is also a power series having the same radius of convergence as f, that is,

A power series is infinitely complex differentiable in its disc of convergence, and the higher derivatives are also power series obtained by termwise differentiation.

Analytic Functions

Definition of Analytic

A function $f$ defined on an open set $Ω ⊂ \mathbb{C}$ is said to be analytic (or have a power series expansion) at a point $z_0 ∈ Ω$ if there exists a power series centered at $z_0$, with positive radius of convergence, such that

for all $z$ in a neighborhood of $z_0$. If f has a power series expansion at every point in $Ω$, we say that $f$ is analytic on $Ω$.

• Useful Remark: The exponential, sine and cosine functions are (by our definition) analytic at 0 and have an infinite radius of convergence. They are automatically defined for all complex numbers.

Analytic and Holomorephic

A holomorphic function is automatically analytic.

Complex Integrals

Definition

• A parametrized curve is a set $\mathcal{C} ⊂ \mathbb{C}$ such that there exists a parametrization

  $$\gamma: I \rightarrow \mathcal{C}$$

  for some interval I → C, where γ is locally injective. We will say that C is smooth if there exists a parametrization γ that is differentiable with $γ′(t) \neq 0$ for all $t ∈ I$.

  Positively and negatively oriented: parametrized in a counter-clockwise and clockwise fashion, respectively.

• Let $Ω ⊂ \mathbb{C}$ be an open set, $f$ holomorphic on $Ω$ and $\mathcal{C^∗} ⊂ Ω$ an oriented smooth curve. We then define the integral of $f$ along $\mathcal{C^∗}$ by

Though the most basic definition should be in the below form, sometimes useful for calculation.

$$\int_{C} f(z) d z=\int_{C}(u+\mathrm{i} v)(d x+\mathrm{i} d y)=\int_{C}(u d x-v d y)+\mathrm{i} \int_{C}(v d x+u d y)$$

• Define the curve length as

  $$\ell(\mathcal{C}):=\left|\int_{\mathcal{C}} d z\right|$$

Basic Property

• $$\int_{-\mathcal{C}^{*}} f(z) d z=-\int_{\mathcal{C}^{*}} f(z) d z$$

• $$\left|\int_{\mathcal{C}^{*}} f(z) d z\right| \leq \ell(\mathcal{C}) \cdot \sup _{z \in \mathcal{C}}|f(z)|$$

Question5

Evaluate the integral, where $C$ the line segment with initial point −1 and final point i; or the arc of the unit circle $Im z ≥ 0$ with initial point −1 and final point i.

Answer5

Independence of Path

If a continuous function f has a primitive $F$ in $Ω$, and $\mathcal{C^∗}$ is a curve in $Ω$ that begins at $w_1$ and ends at $w_2$, then

This is equivalent to

A reminder: Does a horlomorephic function $f$ always have a primitive? Recall $f(z) = 1/z$.

Of course not. A horlomorephic function $f$ defined on an open subset of $\mathbb{C}$ which is also simply connected will have a primitive $F$.

Judgement - Basic

• Goursat’s Theorem:

  Let $Ω ⊂ \mathbb{C}$ be open and $f$ holomorphic on $Ω$. Let $T ⊂ Ω$ be a triangle whose interior is also contained in $Ω$. Then

  $$\oint_{T} f(z) d z=0$$

• Corollary:

  If $f$ is holomorphic in an open set Ω that contains a rectangle R and its interior, then

  $$\oint_{R} f(z) d z=0$$

• Theorem:

  A holomorphic function in an open disc has a primitive in that disc.

• Cauchy’s Theorem:

  If $f$ is holomorphic in a disc, then for any closed curve $\mathcal{C}$ in that disc.

  $$\oint_{\mathcal{C}} f(z) d z=0$$

• Cauchy's Integral Theorem*:

  Let $U$ be an open subset of $\mathbb {C}$ which is simply connected, let $f:U\to \mathbb {C}$ be a holomorphic function, for any closed curve $\mathcal{C}$ in $U$

  $$\oint_{\mathcal{C}} f(z) d z=0$$

• Corollary:

  Suppose $f$ is holomorphic in an open set $Ω ⊂ \mathbb{C}$ containing a circle $\mathcal{C_0}$ and its interior. Then

  $$\oint_{C_0} f(z) d z=0$$

All of the above theorems has one same key point: the existence of primitive in some region, requires there's no "holes" in the region.

Question6

$C$ is the unit circle centered at the origin. Explain, relating to the above theorems, why the below integral does not vanishes to 0. You can draw.

Answer6

Judgement - Toy Contours

• Cauchy’s theorem can be applied to various contours. Below are some toy contours.

  

  Simply means: If $f$ is holomorphic in a contour, then for any closed curve $\mathcal{C}$ in that contour (usually we simply choose the boundary of the contour):

  $$\oint_{c} f(z) d z=0$$

This is acually still a special case for the general Cauchy's Integral Theorem*.

A very useful technieque to eveluate integrations and so on.

Jordan’s Lemma

Assume that for some $R_0 > 0$ the function $g: \mathbb{C} \backslash \overline{B_{R_{0}}(0)}→\mathbb{C}$ isholomorphic. Let

Let

be a semi-circle segment in the upper half-plane and assume that

Then

Cauchy Integral Formulas

Suppose $f$ is a holomorphic function in an open set $Ω ⊂ \mathbb{C}$. If $D$ is an open disc whose boundary is contained in $Ω$, then

where $C = ∂D$ is the (positively oriented) boundary circle of D.

Tricky question: does it matters whether z is in the disk or not? Draw graphs and analysis.

• The values of a holomorphic function within a disc are fixed by the values of the function on the boundary

Tricky reminder: does this means all the values of f(z) in a chosen disk are the same?

• Cauchy’s integral formula is also valid for all of our toy contours

Corollary:

If $f$ is a holomorphic function in an open set $Ω ⊂ \mathbb{C}$, then $f$ has infinitely many complex derivatives in $Ω$. Moreover, if $D$ is an open disc whose boundary is contained in $Ω$,

where $C = ∂D$ is the (positively oriented) boundary circle of $D$.

Question7

Compute $\int_{c} \frac{\mathrm{e}^{z^{2}}}{z-2} d z$ , where C is the curve shown below

Answer7

Question8

Compute $\int_{c} \frac{\mathrm{e}^{z^{2}}}{z-2} d z$ , where C is the curve shown below

Answer8

Question9

Compute $\int_{c} \frac{\mathrm{e}^{z^{2}}}{z-2} d z$ , where C is the curve shown below

Answer9

Question10

Compute $\int_{c} \frac{1}{(z-2)(z-5)} d z$ , where C is the circle with radius 3 and centered at the origin.

Answer10

Holomorphic Functions are Analytic

Suppose $f$ is a holomorphic function in an open set $Ω$. If $D$ is an open disc centered at $z_0$ and whose closure is contained in $Ω$, then f has a power series expansion at $z_0$

for all $z ∈ D$ and the coefficients are given by