@Chen Siyi

November 6, 2020

Midterm2 Part1

Conponents in the Complex Plane

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 $S$.

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

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

Functions in the Complex Plane

Holomorphic Functions

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.

The Cauchy-Riemann Differential Equations

1. If $f$ is holomorphic, then the Cauchy-Riemann equations is satisfied:

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

3. Define two operators:

If $f$ is holomorphic, then

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.

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

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 $t ∈ I$.

  Understand simply, $\gamma$ is parametrizing the "position":

  $$\gamma(t)=x(t)+i y(t)$$

  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.

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

• Define the curve length as

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

Basic Property

• Oriented:

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

• Triangular inequality for integrals:

  $$\left|\int_{\mathcal{C}^{*}} f(z) d z\right| \leq \int_{\mathcal{C}^{*}} \left|f(z)\right| d z$$

  * Triangular inequality:

  $$\left| z_1 \right| - \left| z_2 \right| \leq \left| z_1 + z_2 \right| \leq \left| z_1 \right| + \left| z_2 \right|$$

• Upper bound:

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

Question

Evaluate the integral along two different paths:

1. The line segment with initial point −1 and final point i;
2. The arc of the unit circle $Im z ≥ 0$ with initial point −1 and final point i.

Answer

Cauchy's Integral Theorem

Primitive / Independent of Path

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

This is equivalent to

A holomorphic function $f$ defined in a region $\Omega$ may not always have a primitive. Recall $f(z) = 1/z$.

One way to judge the existence of primitive $F$ is analyzing the region $\Omega$ where the function $f$ is defined.

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 $U$

Specific Cases of Cauchy's Integral Theorem

• 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

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

• Cauchy’s Theorem:

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

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

• Corollary:

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

• Toy Contours:

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

  

  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):

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

Comment on a special case:

All $z^n$ has a primitive except for the case where $n = -1$.

Jordan’s Lemma

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

Let

be a semi-circle segment centered at the origin 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.

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

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

  The reason is actually Cauchy Integral Formulas has a more general way to throw it:

  Suppose $\mathcal{C}$ is a simple closed curve and the function $f(z)$ is holomorphic on a region containing $\mathcal{C}$ and its interior. We assume $\mathcal{C}$ is oriented counterclockwise. Then for any $z_0$ inside $\mathcal{C}$, the integral formula holds. (How do you understand it?)

  

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 $D$.

Question

Compute $\int_{C} \frac{1}{\left(z^{2}+4\right)^{2}} d z$ over the contour shown (using Cauchy's integral formula):

Answer

 

Evaluate Real Integrations

• Extend the real domain to complex domain

  • If only containing x, always directly extend to z
  • If containning sinx, cosx, always extend to $e^{iz}$

• Find poles for the function$f(z)$

• Decide the contour and the branch if needed

  • Semicircle and Indented Semicircle
  • Circle with Keyholes
  • Multiple Kehole
  • Square

• Obtain the complex integral along the whole contour using theorems or formula

  • Cauchy’s integral theorem (no poles contained)
  • Cauchy’s integral formula (one or two poles, not very complicatied)
  • The Residue Theorem (one pole or multiple poles)

• Except for the integral part we need, solve or vanish other parts one by one

  • May need to use Jordan’s Lemma to prove vanishing
  • May need to use triangular inequality and triangular inequality for integrals to prove vanishing

Question

Compute the real integral

$$I=\int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)^{2}} d x$$

Answer

Additional Exercise

*Question

Compute $\int_{C} \frac{z}{\left(z^{2}+4\right)^{2}} d z$ over the contour shown (using cauchy's integral formula):

Answer

Hint:

Apply piecewise integration.

And you can use the residue theorem... (coming soon)