Topology of the Complex Plane, Differentiability, etc

distance between z and w (complex numbers)

$$d\left(z,w\right)=\left|z-w\right|$$

• this “distance function” is used to define limits as it can describe “nearness”

Open disk definition

$$\Delta\left(z_0,\epsilon\right)=\left\lbrace z\in C:\left|z-z_0|<\epsilon\right.\right\rbrace$$

• center at $z_0$

• radius of $\epsilon$

deleted open disk definition

$$\Delta^{\ast}\left(z_0,\epsilon\right)=\Delta\left(z_0,\epsilon\right)\ \lnot\left\lbrace z_0\right\rbrace$$

• center at $z_0$

• radius of $\epsilon$

interior point on A

$A\subseteq C,z_{o}\in C$
we say $z_0$ is an interior point on A if there exists an $\epsilon$ > 0 such that $\Delta\left(z_0,\epsilon\right)\subseteq A$

exterior point of A

we say $z_0$ is an exterior point of A if there exists an $\epsilon>0$

such that $\Delta\left(z_0,\epsilon\right)\subseteq A^{\complement}===\Delta\left(z_0,\epsilon\right)\cap A=0$

boundary point on A

we say $z_0$ is a boundary point on A if for every $\epsilon>0$ we have

$\Delta\left(z_0,\epsilon\right)\cap A\ne0$ AND $\Delta\left(z_0,\epsilon\right)\cap A^{\complement}\ne0$

limit point of A

we say $z_0$ is a limit point of A if, given any $\epsilon$ > 0 we have

$$\Delta^{\ast}\left(z_{0,}\epsilon\right)\cap A\ne\left\lbrace\right\rbrace$$

Open set definition

Let $A\subseteq C$

We say A is open if every point of A is an interior point

closed set definition

Let $A\subseteq C$

We say A is closed if every point of $A^{\complement}$ is an interior point

convergence of a sequence definition (lim)

Let $z_{n}$ be a sequence of complex numbers. We say $z_n$ converges to $z_0$ if, given any $\epsilon>0$ there exists an integer k such that for all $n\ge k$ the distance $\left|z_{n}-z_0\right|<\epsilon$

In that case we write $\lim_{n\to\infty}\left(z_{n}\right)=z_0$

If such a $z_0$ does not exist we say $\lim_{n\to\infty}\left(z_{n}\right)\ DNE$

requirement for limit of function f (deleted open disk)

Let $A\subseteq C$ and let $z_0$ be a limit point of A

Let $f:A\to C$ be a function

We say f has a limit at $z_0$ if there exists a point $w_0\in C$ such that, given any $\epsilon>0$ there exists a $\delta>0$ such that

$$f\left\lbrack A\cap\Delta^{\ast}\left(z_0,\delta\right)\right\rbrack\subseteq\Delta\left(w_0,\epsilon\right)$$

in that case we write $\lim_{z\to z_0}f\left(z\right)=\omega_0$

otherwise DNE

lim f(z) = w_0 <=>

$$\iff\lim_{z\to z_0}|f(z)-w_0|$$

limit and modulus correlary

if $\lim_{z\to z_0}f\left(z\right)=w_0$ then $\lim_{z\to z_0}\left|f\left(z\right)\right|=\left|w_0\right|$

lim[f(z) + g(z)] = ?

$$\lim_{z\to z_0}f\left(z\right)+\lim_{z\to z_0}g\left(z\right)$$

lim[f(z) - g(z)] = ?

$$\lim_{z\to z_0}f\left(z\right)-\lim_{z\to z_0}g\left(z\right)$$

lim [f(z) g(z)] = ?

$$\lim_{z\to z_0}f\left(z\right)\cdot\lim_{z\to z_0}g\left(z\right)$$

lim[f(z)/g(z)] = ?

$$\frac{\lim_{z\to z_0}f\left(z\right)}{\lim_{z\to z_0}g\left(z\right)}$$

Continuous function definition

Let $A\subseteq C$ and let $z_0\in A,f:A\to C$ be a function

We say f is continuous if, at $z_0$ , given any $\epsilon>0$ , there exists a $\delta>0$ such that

$$z\in\Delta\left(z_0,\delta\right)\cap A\Rightarrow f\left(z\right)\in\Delta\left(f\left(z_0\right),\in\right)$$

i.e. f is continuous at $z_0$ if

$$\lim_{z\to z_0}f\left(z\right)=f\left(z_0\right)$$

discontinuous function definition

We say f is discontinuous at $z_0$ if $z_0\in Dom\left(f\right)$ AND either

1) $\lim_{z\to z_0}f\left(z\right)\ DNE$ OR

2) $\lim_{z\to z_0}f\left(z\right)\ne z_0$

continuity properties of two continuous functions

Let f and g be continuous functions at $z_0$ . The following functions are also continuous:

$f\pm g,f\cdot g,\left|f\right|,\overline{f},\operatorname{Re}\left(f\left(z\right)\right),\operatorname{Im}\left(f\left(z\right)\right),\frac{f}{g}$ when $g\left(z_0\right)\ne0$

continuity of a composition of functions definition

Let f and g be functions such that f is continuous at $z_0$ and g is continuous at $f(z_0)$ . Then the function $g(f(z))$ is continuous at $z_0$

where are polynomial functions continuous?

everywhere

Where is Arg(z) continuous?

$$C\lnot\left(-\infty,0\right\rbrack$$

What forms are valid to use LaHopital’s?

$$\frac00,\frac{\infty}{\infty}$$

where is e^z continuous

everywhere on C

Differentiable definition

let $f:A\to C$ where $A\subseteq C$ is an open set

f is differentiable at $z_0$ if

$$\lim_{z\to z_0}\frac{f\left(z\right)-f\left(z_0\right)}{z-z_0}$$

exists

A^n - B^n factoring formula

$$\left(A^{n}-B^{n}\right)=\left(A-B\right)\left(A^{n-1}+A^{n-2}B+A^{n-3}B^2+\cdots+AB^{n-2}+B^{n-1}\right)$$

if f is differentiable at z_0, then it is ____

continuous at $z_0$

critical point for a function f (definition)

$z_0$ is a critical point for f if $f’(z_0)=0$

analytic function definition

a function is analytic (holomorphic or regular) in an open set if it has a derivative at every point in the set

entire definition

a function is entire if it is analytic on the whole complex plane

T or F: if a function is continuous at a point, it is differentiable at that point

False

T or F: if a function is differentiable at a point, it is continuous at that point

True

T or F: if f is not continuous at a point, it can still be differentiable at that point

False

Cauchy Riemann Equations

$$u_{x}\left(x,y\right)=v_{y}\left(x,y\right)$$

$$u_{y}\left(x,y\right)=-v_{x}\left(x,y\right)$$

T or F: if either of the CR equations don’t hold at z_0, the function is not differentiable at z_0

True

T or F: if the CR equations hold at z_0, the function is differentiable at z_0

False

CR-related requirements to ensure f is differentiable at z_0

Let $A\subseteq C$ be an open set and let $z_0=x_0+iy_0\in A$ .

Let $f:A\to C$ be a function such that $u\left(x,y\right)=\operatorname{Re}\left(f\left(z\right)\right),v\left(x,y\right)=\operatorname{Im}\left(f\left(z\right)\right),z=x+iy$

if

1) u and v are $C^1$ functions on some open neighborhood of $z_0$ AND

2) the CR equations are satisfied at $z_0$

then f is differentiable at z_0 and

$$f^{\prime}\left(z_0\right)=u_{x}\left(x_0,y_0\right)+iv_{x}\left(x_0,y_0\right)=\frac{1}{i}\left\lbrack u_{y}\left(x_0,y_0+iv_{y}\left(x_0,y_0\right)\right)\right\rbrack$$

constant function definition using differentiability

Let $f:\left(a,b\right)\to\mathbb{R}$ be a differentiable function. If $f’(x)=0$ for all $x\in(a,b)$ then f is constant on (a,b)

Mean value theorem for real numbers

Let $f:[a,b]\to R$ be a function such that

1) f is continuous on [a,b] AND

2) f is differentiable on (a,b)

then there exists a point $\xi\in\left(a,b\right)$ such that $\frac{f\left(b\right)-f\left(a\right)}{b-a}=f^{\prime}\left(\xi\right)$

$$\Longrightarrow f\left(b\right)-f\left(a\right)=f^{\prime}\left(\xi\right)\left(b-a\right)$$

Does the mean value theorem hold for complex numbers?

Not at all. e^z is an example

What must a set have to be a domain?

it must be open and connected

constant function definition using CR eqns

Let $u(x,y)$ be a real valued function defined on a domain $D\subseteq C$ such that

$$u_x(x,y)=u_y(x,y)=0$$

for all $z=x+iy\in D$ then u is constant on D

constant function definition using analyticity

let $f$ be an analytic function defined on a domain D. If

$f’=0$ for all $z\in D$

then f is constant

branch of an inverse function definition

Let $f:A→C$ be an analytic function and let $D\subseteq f\left(A\right)$ be a domain

A branch of $f^{-1}$ is defined as a continuous function $g:D→A$ such that

$f\left(g\left(z\right)\right)=z$ for all $z\in D$

T or F: a branch of $f^{-1}$ is univalent on the domain

True, always

differentiability of a branch of an inverse function requirements/definition

Let $f:A→C$ be an analytic function and let $g$ be a branch of $f^{-1}$ on domain $D$

Let $z_0\in D$ and let $w_0=g(z_0)$

if $f’(w_0)≠0$

then $g$ is differentiable at $z_0$

Moreover: $g^{\prime}\left(z_0\right)=\frac{1}{f^{\prime}\left(w_0\right)}=\frac{1}{f^{\prime}\left(g\left(z_{}\right)\right)}$

requirement for $g$, a branch of $f^{-1}$ to be analytic

$f^{\prime}\left(w_0\right)\ne0$ for all $w\in g\left(D\right)$

Two most important inverse relations

1) The root relations

2) The logarithmic relations

$p^{th}$ root function branch definition

Let $p\in\mathbb{Z_{\ge2}}$ and let $D$ be a domain. A branch of the $p^{th}$ root function is an analytic function

$g:D→C$ such that

$\left\lbrack g(z)]^{p}\right.=z$ for all $z\in D$

identity (=1) relation of branch of $p^{th}$ root function

$$p\left\lbrack g\left(z\right)\right\rbrack^{p-1}\cdot g^{\prime}\left(z\right)=1$$

Is zero in the domain of a branch of the $p^{th}$ root function?

No, using the identity relation we get 0 = 1.

$g’(z)$ as a branch of the $p^{th}$ root function relation/definition

$$g^{\prime}\left(z\right)=\frac{g\left(z\right)}{pz}$$