Week 1: 1-21: The first day of class was introductory.
The focus of the course will be to understand complex numbers,
functions of complex numbers, and the derivative, integral and series
calculi for complex functions.
Math 418 covers the calculus of complex numbers and complex number valued
functions of one complex variable.
Key theorems we will examine will involve continuity,
differentiability, and integrability. The first coherent and
somewhat successful presentation of calculus for complex functions was by Cauchy in the
early 19th century- about 150 years after the works of Newton and
Leibniz.
We will use a working definition for what a real number is: a real
number is any number that can be represented as a possibly infinite
decimal (positive or negative) or that can be thought of a the
measure of a length of a line segment either to one side or the
other of a specified point on on a given line an a specific unit for
measurement.
Details of the syllabus will be discussed further on Friday.
1-23 [More details to follow.]
The organization of the course:
details like tests, homework, etc. as described in the main
course page.
In the discussion reference was made to Polya's :
How to Solve It...
Polya describes 4 Phases of Problem Solving
1. Understand the problem.
2. See connections to devise a plan.
3. Carry out the plan.
4. Look back. Reflect on the process and results.
It is the first phase that is usually not recognized as being
essential. there is usually more to understand than is apparent.
Also mentioned were other texts in Complex Analysis : See Moodle lists. On Proofs:
Daniel Solow's work was mentioned with on-line references on home page.
How to Read and Do Proofs by D.Solow
The Keys to Advanced Mathematics by D. Solow
Polya's :
How to Solve It... available on line:
https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf
Start work on Complex Numbers:
Reading through class as start on how to: SOS 1.10 (?); 1,12; 1.53; 1.54; 1.55a; 1.56a.;1.59
Algebra and geometry problems from (i) SOS 1.53-1.74 (ii) Aben Chapter 2 (3?) Week 2 1-26: Continue work on Complex Numbers.
Possible pr/exercises to do in class: SOS 1.10 (?); 1,12; 1.53; 1.54; 1.55a; 1.56a.;1.59
Algebra and geometry problems from (i) SOS 1.53-1.74 (ii) Aben Chapter 2 (3?) Class Problem work notes from "secretary's report".:
1.72.
Find an equation for (a) a circle of radius 2 with center at (-3, 4),
(b) an ellipse with foci at (0, 2) and (0, -2) whose major axis has
length 10.
(a) Let $z=x+iy$. We have a radius of 2, center at (-3,4), or $-3+4i$
Cartesian form: $(x+3)^2+(y-4)^2=4$
But this doesn't really answer this question the way we want it to...
Complex form: if the center is at $-3+4i$, and we want $r=2$, we are
saying that the distance away from $z$ to $3+4i$ is $2$. Recall that modulus of
$z$, $|z|$ measures distance/length of vector.
So, the equation is $|z-(-3+4i)|=2$
(b) foci (0,2) and (0,-2) are $2i$ and $-2i$, respectively
Recall
from the definition of an ellipse that the total of the distances from the foci
to a point on the ellipse will be equal to the length of the major axis.
So, $|z-(-2i)|+|z-2i|=10$
Note:
$|z|$ is a function that takes a complex number to a real numbers and
is not one to one. (Two different z values will map to the same real
number.)
Look at technology for visualizing complex numbers: Geogebra -
1-28: Continue foundational work on Complex Numbers.
Start on how to: SOS 1.7(a); 1.19; 1,12; 1.22;1.43; 1.44; 1.55a; 1.56a.;1.59
Algebra and geometry problems from (i) SOS 1.75-1.99 (ii) Aben Chapter 2 (3?) CORRECTION/NOTE from discussion of 1-26 : $ |z|^2 = z \cdot \overline z$.where $\overline {a+bi} = a - bi$. Notes from "secretary's report":
1.82 Show that $2+i = e^{i \arctan(\frac 12)}$
Solution: If
$ z = a+bi$, we know that $|z| = \sqrt {a^2 + b^2}$. Also, we know that $\theta =
\arctan(\frac ba)$. Lastly, we know $z = a+bi = |z|e^{i \theta}$. Hence $|z| =
\sqrt{2^2 + 1^2} = \sqrt(5); \theta = \arctan(\frac 12)$. Therefore $z = 2+i = \sqrt 5 e^{i \arctan(\frac 12)}$.
We also "proved" Euler's formula, $e^{i \theta} = \cos(\theta)+i \sin( \theta)$, by
using the taylor series expansion of all three terms.
Since all three
series are absolutely convergent, we can rearrange the left side in such
a
way that $e^{i \theta} = \sum_{n=0}^{\infty} \frac{(i
\theta)^n}{n!} = \sum_{k=0}^{\infty} \frac{(-1)^k
\theta^{2k}}{(2k)!} + \sum_{k=0}^{\infty} \frac{(-1)^k i
\theta^{2k+1}}{(2k+1)!} = \sum_{k=0}^{\infty} \frac{(-1)^k
\theta^{2k}}{(2k)!} +
i \sum_{k=0}^{\infty} \frac{(-1)^k \theta^{2k+1}}{(2k+1)!}
= \cos(
\theta)+i \sin( \theta)$.
1.97 a) Find the roots of $z^4+81 = 0$.
Solution:
Note if $z$ is a root of the equation then $z^4=-81 = 81 e^{\pi i}$.
Using Euler's formula with $z = re^{i \theta}$, we have $r^4 = 81$,
hence $r = 3$. Also using
$z^n
= r^n(cos( (n \theta ) +isin( (n \theta ))$ with $n = 4$ we use the periodicity of the sine and cosine to conclude that
$4\theta = \pi, 3\pi, 5\pi,7\pi$. Therefore by solving and plugging
the $\theta$
values back into the sine and cosine functions we obtain the roots
are $z =3(\pm \frac{\sqrt 2}2 \pm \frac {\sqrt 2}2 i )$.
1-30 Notes from "secretary's report".
1.88 a) Prove that $r_1e^{i \theta_1} + r_2e^{i \theta_2} = r_3e^{i \theta_3}$
where
$r_3= \sqrt{r_1^2+r_2^2+2r_1r_2cos( \theta_1 - \theta_2)} $
and $\theta_3= \arctan(\frac{r_1sin( \theta_1)+r_2sin(
\theta_2)}{r_1cos( \theta_1)+r_2cos( \theta_2}))$.
Solution: We
know that a complex number, $a+bi$ can be written in the form $
re^{i \theta}$ where $ r = \sqrt{a^2+b^2}$ and $\theta = |arctan (\frac ba)$. If
we give ourselves 2 arbitrary complex numbers, $z_1=a_1+b_1i $and $ z_2= a_2+b_2i$,
we can say that the sum of two complex numbers amounts to complex
number $z_3= a_1+b_1i +a_2+b_2i= (a_1+a_2) +i(b_1+b_2)$.
Calculating the radius for $z_3$, we have $r_3 = \sqrt{(a_1+a_2)^2 +(b_1+b_2)^2)}$.
Recognizing
the "dot product" of the vectors $<a_1,b_1>\cdot<a_2,b_2> =
a_1a_2+b_1b_2 $, We see that $a_1a_2+b_1b_2 = r_1r_2cos( \theta_1
- \theta_2)$, and so substituting
into our modulus for $z_3$, we have $r_3= \sqrt{r_1^2+r_2^2+2r_1r_2cos(
\theta_1 - \theta_2)} $ as desired.
To
show that $\theta_3= arctan(((r_1)sin( \theta_1)+(r_2)sin(
\theta_2))/((r_1)cos( \theta_1)+(r_2)cos( \theta_2)))$, we know that
imaginary part for $z_3$ is given by $r_1sin( \theta_1)+ r_2sin(
\theta_2)$.
The
real part of $z_3$ is given by the sum of $r_1cos( \theta_1)$ and
$r_2cos( \theta_2)$. Taking the arctangent of these two components yields the
desired answer, $\theta_3= \arctan(\frac{r_1sin( \theta_1)+r_2sin( \theta_2)}{r_1cos( \theta_1)+r_2cos( \theta_2}))$.
Later in the class, a student posed a question: what is the value of $i^i$? Turns out it’s a real number(no pun intended)!
Solution: We use the fact that $i = e^{i \frac {\pi}2}$
We consider $i^i$ by applying exponent rules and we have $i^i = {(e^{i \frac{\pi}2 }})^i =e^{i\cdot i\frac{\pi}2}$
so.... $i^i = e^{-\frac{\pi}2} \approx 0.2078$. This was verified using GeoGebra.
Week 3
2-2: Solving Polynomial Equations-The Rational Root Theorem works for complex with polynomials with integer coefficients:
Take the ratio of the factors of the first and last coefficients and
proceed by trial and error. Long division is require.
Problem
1.101: Find the roots of $p(z)=z^5-2z^4-z^3+6z-4$. Solution: Using the rational
root test, the possible roots are ±1, ±2, ±4. Using substitution, we
know that $p(1)=p(2)=0$.
So $q(z)=(z-1)(z-2)=z^2-3z+2$. Now we divide $p(z)$ by $q(z)$ to get $r(z)=z^3+z^2-2$. Hence,
$r(z)∙q(z)=p(z)$, and we have to find the roots of $r(z)$, which are $1, -1+i$ and $-1-i$.
Region of complex inequality. In particular we looked at the “annulus” or “ring”.
$R_1<|z-(a+bi)|<R_2$ centered at $ a+bi$.
$1<|z+i|≤2$ is the region below in between the dashed-line circle and the solid-line circle.
Proving trigonometric identities using De Moivre’s theorem
Problem 1.93. Prove $\frac {\sin (4\theta)}{\sin (\theta)}= 8\cos^3(\theta) - 4 \cos(\theta) = 2\ cos(3\theta) + 2 \cos(\theta)$ Idea: $\sin(4|theta) = IM\{ ( \cos(\theta) + i \sin(\theta))^4\}$. This
turns out to have a common factor of $\sin(\theta)$.
2-6: Geometric interpretation of complex conjugate. $ z = x+ yi$ , $\overline z = x - yi$. $\overline z$ is denoted in the figure by $z^*$.
It's a reflection over the Real Axis
Problem 1.133. Let $P(z)$ be any polynomial with real coefficients. Prove $\overline {P(z)} =P( \overline z)$ assuming coefficients are real. Idea: Use the facts that $\overline{z\cdot w} = \overline z \cdot
\overline w$, $\overline{z + w} = \overline z+\overline w$ and $\overline a =
a$ when $a$ is a real number.
$a+bi = \left( \begin{array}{cc}
a & b \\
-b & a \end{array} \right) \\ |a +bi| = det \left( \begin{array}{cc}
a & b \\ -b & a \end{array} \right) $
Problem 2.50 (d) $f(z) =\frac{ z+2}{2z-1}$. Find values of $z$ such that $f(z) = z$. Solution: $\frac{ z+2}{2z-1} = z$ so $z+2 = z \cdot (2z-1) = 2z^2 -z$.
So $2z^2 -2z -2 = 0$ or $z^2 -z -1 = 0$ and by the quadratic
formula $ z =\frac{ 1 \pm \sqrt{1 + 4 }}2 =\frac{ 1 \pm \sqrt 5}2 $. Checked with winplot:
Week 4
February 9th - 11th
On February 9th and early February 11th we worked on an example from the
text and studied some of the properties of the function provided using
the interactive complex mapping diagram.
The function we studied is as follows:
After smearing a little complex algebra around, we attained the real and
complex parts and expressed them as functions of $x$ and $y$ as
follows:
$f(z) = x(2 - x) + y^2 + i((2y - xy) - xy)$
which yields
$u(x,y) = 2x - x^2 + y^2$ and $v(x,y) = 2y - 2xy$
Surprisingly, the real and imaginary parts of $f(z)$ satisfy the
Cauchy-Riemman Condition which is a necessary condition for
differentiability of complex functions. Observe:
You may study the function's properties using the complex mapping diagram found at:
We then looked at an example from the SOS pr/exercises where we showed the following:
All of the roots of $cos(z) = a$, where $-1 \leq a \leq 1$, are real.
Our immediate goal was to show that $Im\{ z \} = 0$.
This was done by expressing $cos(z)$ in terms of exponentials so:
Assume $cos(z) = a$, where $-1 \leq a \leq 1$. Then
$cos(z) = \frac{e^{iz} + e^{-iz}}{2}= a$
$\iff \frac{e^{iz} + e^{-iz}}{2} = a$
$\iff e^{iz} + e^{-iz} = 2a$
$\iff e^{iz} + \frac{1}{e^{iz}} = 2a$
$\iff(e^{iz})^2 + 1 = 2a e^{iz}$
$\iff(e^{iz})^2 - 2a(e^{iz}) + 1 = 0$
$\Rightarrow e^{iz} = a \pm \sqrt{a^2 -1}$
$\iff e^{-y}e^{ix} = a \pm i\sqrt{1 - a^2}$ assuming $z = x + iy$
$\Rightarrow|e^{-y}e^{ix}| = \sqrt{a^2 + 1 - a^2}$
$\iff e^{-y} = \sqrt1 = 1$ which happens only when $y = 0$
Thus $Im\{ z \} = 0$, and $z = x$, a real number.
Thus all of the roots of $cos(z) = a$, where $-1 \leq a \leq 1$, are real. We ended lecture on Wednesday studying the geometric properties of basic functions on the $W$ - plane.
Using Winplot we looked at the most basic function:
$w = f(z) = z +a + ib $
Animating on $a$ and keeping $b$ constant makes the image of any point
on the $z$ - plane shift either left or right depending on the sign of
$a$. Similarly, animating on $b$ and keeping $a$ constant makes the
image of any point on the $z$ - plane shift either up or down depending
on the sign of $b$.
We next looked at the function:
$w = f(z) = z \cdot (a + ib)= z \cdot\rho e^{\theta i}$
Animating on $\rho$ "magnifies" the image of a set of points on the $z$ -
plane, while animating on $\theta$ rotates the image of a set of points
on the $z$ - plane depending on the angle $\theta$ of $(a + ib)$. 2-13
Note that $\frac{du}{dy}$ and $\frac{dv}{dx}$ are of opposite sign, but this is not always the case.
F$(z) = z = x-iy$
$\frac{du}{dx} = 1$
$\frac{du}{dy} = 0$
$\frac{dv}{dx} = 0$
$\frac{dv}{dy} = -1$ We then discussed the
general way that a derivative is found with linear functions, and how
this relates to finding derivatives of complex functions.
Linear function example for a real valued function:
$f(2) = 5$, $f(3) = 7$
What's the slope?
$\frac{f(3)-f(2)}{3-2} = \frac{7-5}{1} = 2$
$f(x) = 7+2(x-3) = 7+2x-6 = 2x+1$ Complex linear function example:
$f(1+i) = 2+i$, $f(1-i) = 1+i$
"Slope": $\frac{f(1+i)-f(1-i)}{(1+i)-(1-i)} = \frac{(2+i)-(1+i)}{1+i-1+i} = \frac{1}{2i} = \frac{-1}{2}i$
$f(z) = (2+i)+ \frac{-1}{2}i(z-(1+i)) = 2+i-\frac{1}{2}iz+\frac{1}{2}i-\frac{1}{2} = \frac{3}{2}+\frac{3}{2}i-\frac{1}{2}iz$
Check: $f(1-i) = 2+i+\frac{-1}{2}i(1-i-1-i) = 2+i-\frac{1}{2}i(-2i) = 2+i-1 = 1+i$
The term $\frac{-1}{2}iz$ serves as the rotation
factor. Any vector z is first cut in half and rotated 90 degrees
clockwise around the point $2+i$. We then went to
An example: Using the definition of complex number derivatives for the function $f(z) = z^3$,
find $f'(z)$ when $z = 1+i$
Step 1: Evaluate
$f(z) = z^3$
$f(1+i) = (1+i)^3$
$f(1+i)$ has a length of $\sqrt{2}$ and angle
$\frac{\pi}{4}$, while $(1+i)^3$ has length $\sqrt{2^3}$ and angle
$\frac{3\pi}{4}$.
Step 2: Find the difference of the function values at $z$ and $1+i$.
$f(z)-f(1+i) = z^3-(1+i)^3$
Step 3: Divide:
$\frac{f(z)-f(1+i)}{z-(1+i)} = \frac{z^3-(1+i)^3}{z-(1+i)} = z^2+z(1+i)+(1+i)^2$, where $z \neq (1+i)$.
This solution holds from the rule of factoring cubic
polynomials for real number polynomials, since this rule works the same
for complex number polynomials
This is because the complex numbers and the real numbers are both "fields"..
Step 4: Think
Think about $z$ approaching $1+i$
As $z \to 1+i$ , we see that
$\frac{f(z)-f(1+i)}{z-(1+i)}= z^2+z(1+i)+(1+i)^2 \to 3(1+i)^2 = f'(1+i)$
There are two equivalent ways to define the complex derivative of
$w=f(z)$ at $z=z_0$, denoted $f'(z_0)$ or $\frac {dw}{dz}|_{z=z_0}$:
1: $\lim_{z → z_0} \frac{f(z)-f(z_0)}{z-z_0} = f'(z_0)$
2: $z-z_0 = \Delta z, \lim_{\Delta z \to 0} \frac{f(z_0+\Delta z)-f(z_0)}{\Delta z} = f'(z_0)$
Week 5
Notes: 2/16/2015 and 2/18/2015
Suppose $f(z)=u(x,y)+iv(x,y)$
$lim_{∆z →0}\frac{f(z+ ∆z)-f(z)}{∆z}$ ; Assumption: $f(z)$ has a derivative at $z$.
Assume $∆y=0$ then: $lim_{∆x→0}\frac{u(x+ ∆x,y)+iv(x+ ∆x,y)-u(x,y)-iv(x,y)}{∆x}$
$= ∂u/∂x+i ∂v/∂x$
Assume $∆x=0$: $lim_{∆y→0}\frac{u(x,y+ ∆y)+iv(x,y+ ∆y)-u(x,y)-iv(x,y)}{i∆y}$
$ = 1/i(∂u/∂y+i ∂v/∂y)$
$= (i/i)(1/i)(∂u/∂y+i ∂v/∂y)$
$= ∂v/∂y-i ∂u/∂y$. Equate the real and imaginary components and we have the Cauchy-Riemann Equations! $∂u/∂x= ∂v/∂y , ∂u/∂y= -∂v/∂x$ Example: $f(z)= 3z$,when $z\ne1$
$f(z)= 7$, when $z=1$.
Limits: we don’t care what actually happens at the point $z_0,$ only when near to $z_0$.
Notation: $D_δ={ z∶0< |z-1|< δ}$
Goal: be close to the target number
Find a way to describe $f(z)$ being close to $3$
Example: How do we make $f(z)$ Within $0.1$ of $3$.
|$f(z)- 3|< 0.1$
Find a set of $z$'s in the domain that satisfies closeness to $1$ ($\ne 1$)
We want $|3z- 3|< 0.1$.
or $|3(z- 1)|< 0.1$
that is, $3|z- 1|< 0.1$
so we want$ z-1|<\frac{0.1}3 ≈.03333333333333333333…$
Choose $\delta = 0.03$
Then if $0<|z- 1|< \delta$, then $f(z) – 3 = 3(z-1)$
So $|f(z)- 3|=3|z- 1|<3\delta = 0.09< 0.1$.
If you wanted to be within $0.01$ of $3$, I can find a deleted neighborhood of 1 with $\delta = 0.003$ where
if $0<|z- 1|< \delta$, then $f(z) – 3 = 3(z-1)$
So $|f(z)- 3|=3|z- 1|<3\delta = 0.009< 0.01$.
What about a tiny $\epsilon >0$?
Find a way to describe $f(z)$ being close to $3$ within $ε$ of $3$.
Given any $ε>0$-- so ...
Choose any $ε*>0$ (a fixed, specific $\epsilon$.)
Let $\delta = \frac{ε*}3$.
Then consider $z$ where $0< |z- 1|< \delta$.
Then $f(z) = 3z$
$f(z) – 3 = 3z – 3$
$ = 3(z-1)$
and sp $|f(z)- 3|=3|z- 1|<3\delta = 3 \cdot \frac{\epsilon *}3< \epsilon *$.
So $\lim_{z \to 1}f(z)= 3$. 2-20 SOS: Problem 2.89 (a) Show $lim_{z \to i} z^2+2z = 2i-1$. Solution: Suppose $ \epsilon >0$.
We need to find $\delta >0$ so that if $0<|z-i|<\delta$ then $|z^2+2z -(2i-1)| <\epsilon$.
Note that $|z^2+2z -(2i-1)| = |z^2- i^2 +2z -2i| = |(z+i)(z-i) +2 (z-i)| =|z+i+2||z-i|< |z+i+2| \delta$.
Since we can choose $\delta<1$ ,
we can show using the triangle inequality that $|z+i+2|=|z-i+2i+2| \le
|z-i|+|2i+2| <\delta + 3<4$. So $|z^2+2z -(2i-1)| < 4\delta$.
So we want $ \delta$ so that $4\delta < \epsilon$.
So choose $\delta = min\{ \epsilon/4 ,1/2\}$, and we have
$|z^2+2z -(2i-1)|< 4\delta \le 4 \epsilon/4 = \epsilon$.
Check with some GeoGebra examples on http://web.geogebra.org/app/?id=TBwX2yZs Other discussion:
Consider $f(z) = \frac 1{z-i}$ and the definition of a "domain" for the
complex plane as an open and connected set of complex numbers.
The conventional domain of this function would be $D = \{z : z \ne i\}$
A set $O$ is defined as "open" if and only if for any $z_0 \in O$ there is a $delta \gt 0$ where$\{z:|z- z_0| \lt \delta \} \subset O$
With this definition one can show that $D$ is an open set.
Outline: Choose $z_0 \in D$. let $\delta = \frac{|z_0 -i|}2$. Then it
can be shown that $i \notin \{z:|z- z_0| \lt \delta \}$ we have $\{z:|z-
z_0| \lt \delta \} \subset D$ which shows $D$ is an open set.
For the concept of connected we can use a provisional definition that an open set $O$ is connected if given any two numbers in $O$, $w_1$ and $zw2$, there is a set of line segments in $O$ that "connect" the two numbers.
It was illustrated how the set $D$ is connected.
Proposition: If $S = \{ z_1, z_2, ...z_n\}$ is a finite set of complex numbers then $D = C - S$ is an open connected set. Proof Plan: For $z_0 \in D$ let $\delta = \frac 12 \min \{|z_0-z_k|, k = 1, ... ,n\}$.
For any two numbers in $D$, $w_1$ and $w_2$ ,there is a set of line
segments that "connect" the two numbers while avoiding $S$, so the
segments are in $D$.
Final example: The set $S = \{z : z \ne \frac ik, k = 1,2,3,...\}$ is not an open set.
Discussion: Consider $z=0 \in S$. For any $\delta \gt 0$ there is a
number $k* \gt \frac 1{ \delta}$ so $|\frac ik | = \frac 1k \lt
\delta$ and thus there is no $\delta \gt 0$ where $\{z:|z-0| \lt \delta
\} \subset S$. Week 6
Topology/Geometry of the Domain
Definitions Open Sets: A set $O$ is open if (and only if)
for all $z_0 \in O$, there is a number $\delta > 0$ so that
$\{ z :| z - z_0| < \delta \}\subset O$. Closed Sets: A set $S$ is closed
if the complementary set $S^c = \{z : z \notin S \}$ is an
open set. In
other words: The set $S$ is closed if for any $z_0 \notin
S$, there is a number $\delta > 0$ so that $\{ z :| z - z_0|
< \delta \}\subset S^c$. Stated in a direct way: $S$ is a closed set
if (and only if) whenever $z_0 \in S$ and $\delta > 0$
either $\{ z :| z - z_0| < \delta \}\subset S $ or for any number
$\delta > 0$ , $\{ z :| z - z_0| < \delta \}$ contains an
element $z_* \in S$ and
$z' \notin S$.
Connected Sets. An open set S is said to be connected if any two
points of the set can be joined by a path consisting of straight line
segments (i.e., a polygonal path) all points of which are in S.
Bounded Sets: The magnitude of all elements of a bounded set are smaller or larger than some given number. Geometrically:
Boundaries: The boundary of a set, $S$, denoted $\partial S$ is
given by those points which are surrounded by both points within and
outside of the set, that is, $z_0 \in \partial S$
if and only if for any number $\delta > 0$ , $\{ z :| z - z_0|
< \delta \}$ contains an element $z_* \in S$ and $z' \notin
S$.
Compact Set: A compact set is a set that is both closed and bounded.
Functions
2 main types of functions:
● Linear; of the form $f(z) = az + b$ where $a,b ϵ C $
● Mobius transformation; of the form $f(z) =\frac {az+b}{cz+d}$ where $a,b,c,d ϵ C$.
Functions transform the complex $z$ plane to the complex $w$ plane, where $w = f(z)$.
Three main notions about functions: Limits, Continuity, and Differentiability Limits
The number $L$ is the limit of $f(z)$ as $z$ approaches $z_0$, written
as $\lim_{z→z_0} f(z) = L$, if for some small positive number $\epsilon
\gt 0$ we can find some small positive number $\delta \gt 0$ such
that $|𝑓(𝑧)−𝐿 |<\epsilon$ whenever $0 < |𝑧−𝑧_0| \lt
\delta$..
The limit must be independent of the manner in which $z$ approaches $z_0$.
Continuity
A function is said to be continuous at $z_0$ if $\lim_{z→z_0}f(z) = f(z_0)$. This implies the following three conditions:
1. $lim_{z→z_0} f(z) = L$ must exist
2. $f(z_0)$ must exist
3. $L = f(z_0)$.
Derivatives
The derivative of a single valued function, $f(z)$, in some domain of the complex $z$ plane is defined by:
$𝑓′(𝑧) = 𝑙𝑖𝑚_{\Delta z→0}]\frac { 𝑓(𝑧+𝛥𝑧)−𝑓(𝑧)}{ 𝛥𝑧 } $
provided that the limit exists independent of the manner in which $∆z$
goes to $0$. $f$ is said to be differentiable at $z$ if this is the
case.
A necessary condition for $f(z)$ to be differentiable at all points in a region is to satisfy the Cauchy-Riemann equations:
where $u$ and $v$ are respectively the real and imaginary parts of the transformation $f(z)$.
The Real Projective Line:
Every point on the real number line gets associated with a point on a
unit circle. -infinity and +infinity get mapped to the “north” pole of
the circle which is refer as to the point at infinity. Visually:
The analog of the Cartesian plane for the Real Projective line is the torus.
A similar construction attaches a single point at infinity to the complex numbers creating the "Riemann sphere."
2-27
Compactness in the Complex plane.
Definition: A subset $K$ of the complex numbers is called compact if $K$ is closed and bounded. Theorem : Heine- Borel. (Not proven here- see a real analysis or advance calculus course, e.g Calculus on Manifolds by M. Spivak.)
$K$ is a compact set if and only if for any family of open sets $\{U_{\alpha}\}$ with $\alpha \in I$, $ I$ an indexing set for the family of sets, with the property that $K \subset \bigcup U_{\alpha} $.
then there is a finite collection of the $
U_{\alpha}$, which we can denote $\{ U_1, U_2, U_3, ..., U_n\}$ where
$K \subset \bigcup U_j $.
Compactness and continuous functions.
Theorem: Suppose $f$ is a continuous function on a domain $D$ and $K$ is a compact subset of $D$. Let $K' = f(K)$. Then $K'$ is compact. Proof: Basic outline using the topological characterization of compact related to open covers.
Suppose $U'_{\alpha}$ is a family of open sets with $\alpha \in I$, $ I$
an indexing set for the family of sets, where $K' \subset \bigcup
U'_{\alpha}$.
Let $U_{\alpha}= f^{-1}(U'_{\alpha})$.
Since $f$ is a continuous function, each $U_{\alpha}$ is an open set and $K \subset \bigcup U_{\alpha}$.
Now since $K$ is compact, there is a finite collection of the $
U_{\alpha}$, which we can denote $\{ U_1, U_2, U_3, ..., U_n\}$ where $K
\subset \bigcup U_j$.
Now consider the corresponding family of sets, $\{ U'_1, U'_2, U'_3, ..., U'_n\}$.
For any $w' \in K'$, $w' = f(z)$ for some $z \in K$. But we have $z \in
U_j$ for some $j$, so $w' = f(z) \in U'_j$. Thus if $w' \in K' = f(K)$,
then $w' \in \bigcup U_j '$, so $K' \subset \bigcup U'_j$ for a finite
number of the original $U'_{\alpha}$. Thus $K'$ has been shown to be
compact. EOP.
Week 7 [ Notes are missing :( ] Week 8 3-09
We began the lecture today by observing the following properties:
$e^{\pi i} = cos \pi + i sin \pi = -1$
$e^{-\pi i} = cos (-\pi) + i sin (-\pi) = -1$
so, $log(-1) = ln(-1) = \pi i$ or $-\pi i$
where $ln(z)$ is not continuous at any negative real values.
Observe:
Given $w = e^{x+iy}$, $ln(w) = x + iy + 2\pi ki$, where $2\pi ki$ defines periodicity.
We looked at two ways to interpret moving from $e \to -1$ in the $w$
plane : counterclockwise through upper half plane or clockwise
through the lower half plane.
$e \to -1$ in the image counterclockwise goes from $1 \to \pi i$ in the domain.
$e \to -1$ in the image clockwise goes to $1 \to -\pi i$ in the domain.
The mapping rotates around from $1$ to $\pi i$ and from $1$ to $-\pi i$
forming a continuous sheet (similar to $arctan(z)$ at $z =
\frac{-\pi}{2}, \frac{\pi}{2}$)
(Flashman may want to add a little extra here concerning the spiraling Riemann surface [see solved exercise 2.15]).
Multi-valued function with sheets is continuous, single-valued (just $1$ to $\pi i$ and $1$ to $-\pi$ i) is not?
3-11
Today we looked at the following mapping:
$w=(z) = \frac{3+z}{2-z}$
Observe that the function is doing some inversion.
Note that at $z = 0, 1, i$, and $\infty$
$f(0) = \frac{3}{2}$, $f(1) = 4$
$f(i) = \frac{3+i}{2-i} = \frac{(3+i)(2+i)}{5} = \frac{5+5i}{5} = 1+i$, and
$\lim_{z \to \infty}f(z) = \lim_{z \to 0} f(\frac{1}{z}) = \lim_{z \to
0} \frac{3+\frac{1}{z}}{2-\frac{1}{z}} = \lim_{z \to
0} \frac{3z+1}{2z-1} = -1$.
We then began looking at bilinear transformations.
(note: A more thorough explanation of these transformations can be found on Section 8.10 pg 245 of our text.)
A bilinear transformation is defined as $w = f(z) = \frac{\alpha
z+\beta}{\gamma z+\delta}$ with $\alpha\delta - \beta\gamma \neq 0$
Observe that the transformation is a combination of rotation, translation, stretching, and inversion.
If $z_1$ through $z_4$ are distinct points in the $z$ plane, then
$\frac{(z_4 - z_1)(z_2 - z_3)}{(z_2 - z_1)(z_4 - z_3)}$ is invariant
under the mapping.
(note: invariant means that transformation by $w$ results in the same complex number).
We then looked at some of the properties of this bilinear transformation
as layed out by the solutions to solved problems 8.14 and 8.15 in our
text.
The inversion transformation (factor) changes a circle to either a circle or a
line if a $z$ plane circle passes through a point where the denominator
is zero.
Ex. A circle passing through $(2,0)$ [ such as $|z-1|
= 1$ ] will be transformed by the bilininear transformation
$w=(z) = \frac{3+z}{2-z}$ into a line.
A point in the interior of a circle that has $(2,0)$ in its interior in
the $z$ plane will be sent to the exterior of the transformed circle in
the $w$ plane.
3-13
For a more detailed approach to today's lecture read the section
"Projective Matrix Representations" at the Mobius Transformation
wikipedia page. The link can be found on moodle.
Homogeneous Coordinates - A 2 dimensional (column) vector can represent a real number.
[Think of fractions as ordered pairs.]
$ a = \left( \begin{array}{cc}a \\ 1
\end{array} \right) \tilde \left( \begin{array}{cc} ta \\ t \end{array} \right)$
Comment: There is a similar approach for
complex numbers as ordered pairs of real numbers represented as 3
dimensional column vectors.
$ ( 3,4) = \left( \begin{array}{cc}3 \\ 4 \\1
\end{array} \right) \tilde \left( \begin{array}{cc} 6 \\
8 \\ 2 \end{array} \right) \tilde \left(
\begin{array}{cc} 3t \\ 4t \\ t \end{array} \right)$
An ordinary complex number can be represented likewise as a 2 dimensional vector:
$ \left( \begin{array}{cc}a \\ b
\end{array} \right) \tilde \left( \begin{array}{cc} \frac ab\\
1 \end{array} \right)$ as long as $b \ne 0$, but if $ b =0$ then
$ \left( \begin{array}{cc}a \\ 0
\end{array} \right) $ corresponds to $\infty$. Bilinear Transformations as matrix multiplication:
Consider
$w = f(z) = \frac{\alpha z+\beta}{\gamma z+\delta}$ and the matrix : $T =
\left( \begin{array}{cc} \alpha & \beta \\ \gamma
& \delta
\end{array} \right)$.
Then $ \left( \begin{array}{cc} \alpha & \beta \\
\gamma
& \delta
\end{array} \right)$ $ \left( \begin{array}{cc} z \\ 1
\end{array} \right) = $$ \left( \begin{array}{cc}\alpha z+\beta
\\ \gamma z+\delta \end{array} \right) $ $\tilde
\left( \begin{array}{cc}\frac{\alpha z+\beta}{\gamma z+\delta} \\
1 \end{array} \right)$
The determinant of the matrix $ \left|
\begin{array}{cc}\alpha & \beta \\\gamma
& \delta
\end{array} \right| ≠ 0$ by assumption, thus the matrix is nvertible and
its inverse matrix represents its inverse bilinear trasnformation.
Note: $\det(T^{-1}) = \frac 1{\det(T)} \ne 0$.
Note that with this representation we can see that the
transformation with matrix $T$ is the same transformation as that
represented by the matrix $\alpha T$ where $ \alpha ≠ 0$ since $(\alpha
T)v = \alpha(Tv) ~ Tv$.
What happens at infinity?
Example: $w = f(z) = \frac{3 z+5}{-i z+ 2i}$ with matrix $T = \left( \begin{array}{cc} 3 & 5\\ -i
& 2i
\end{array} \right)$:
Then $ \left( \begin{array}{cc}3 & 5\\ -i
& 2i
\end{array} \right)$ $ \left( \begin{array}{cc} 1 \\ 0 \end{array}
\right) = $$ \left( \begin{array}{cc}3 \\ -i \end{array} \right) $ $\tilde \left(
\begin{array}{cc}\frac{3}{-i} \\ 1
\end{array} \right) $, so $f(\infty) = 3i$.
This is consistent with $\lim_{z \to \infty} \frac{3 z+5}{-i z+ 2i} = \frac 3{-i}$
Examples: A simple (linear) example: $T$ such that $T(0) = 3, T(1) = i, T(\infty) = \infty$
Comment: Here, Since $T$ is linear in this example, $\gamma = 0$. If
$\gamma ≠ 0$ then all coefficients can be divided by $\gamma$, [so we
can assume $\gamma = 1$ andand the pole will be at
$-\delta/\gamma$ instead of at $\infty$
Example: Consider a bilinear transformation $T$ such that $T(0) = 1, T(1) = -i, T(i) = 3$.
$ \left( \begin{array}{cc} \alpha & \beta \\ \gamma
& \delta
\end{array} \right)$ $ \left( \begin{array}{cc} 0 & 1 & i \\ 1 & 1 & 1 \end{array}
\right) =\left( \begin{array}{cc} \beta & \alpha+\beta &
\alpha i + \beta\\ \delta & \gamma+\delta & \gamma i +\delta
\end{array} \right) = $ $ \left( \begin{array}{cc}\ t & -iu & 3v \\ t & u& v \end{array} \right) $
So here are some immediate consequences of this matrix equation that can
lead to solving for a matrix representative of $T$: [6 linear equations
in 7 unknowns.]
$ \beta=t, \delta=t; \beta=\delta$;
$\alpha i+\beta=3v, \gamma i+\delta=v; \alpha
i+\beta=3(\gamma i+\delta)$
$ \alpha+\beta=-iu,
\gamma+\delta=u; \alpha+\beta=-i(\gamma+\delta)$
Solving these equations can be done using linear algebra.. 3-23: The Definition of the line integral for complex functions.
Recall the definition of the integral for real valued functions from beginning calculus:
Suppose $P: [a,b]\to (-\infty. \infty)$. Let $N \gt 0$ be a
natural number and $\Delta x=\frac {b-a}N$ so that
$x_0=a, x_k=a+k\Delta x, x_N=b$; and $x_{k-1}
\le c_k
\le
x_k,
k=1,2,3,... N$.
We let $S_N=P(c_1)\Delta
x+P(c_2)\Delta
x+...+P(c_N)\Delta x = \sum_{k=1}^N P(c_k)\Delta x$.
So that as $N \to \infty$ and $\Delta x \to 0$, we have
$S_N \to \int_a^b P(x)dx = F(b) - F(a)$ where $F'(x) =P(x)$.
We have a comparable definition for the line integral of a complex
function $P: \mathbb{C} \to \mathbb{C}$ over a curve $C$ parametrized by
the function $\gamma = z: [a,b] \to \mathbb{C}$.
Let $N \gt 0$ be a natural number and $\Delta t=\frac {b-a}N$ so that
$t_0=a, t_k=a+k\Delta t, t_N=b$; $t_{k-1}
≤ t_k
≤ t_k$, AND $\Delta z_k = z (t_k) - z (t_{k-1})$ for $k=1,2,3,... N$
We let $S_N=P( z (c_1))\Delta z_1+P( z (c_2))\Delta z_2+...+ P( z
(c_N))\Delta z_N \\ = \sum_{k=1}^N P( z (c_k))\Delta z_k$
So that as $N \to \infty$ and $\Delta z_k \to 0$, and we have
$S_N \to \int_C P(z)dz = \int_a^b P( z(t)) z '(t) dt$.
E.g. $w=P(z)=z^2$ $C: z(t)=t+2ti, 0≤ t ≤1
$ $C$ is a line segment from $z=0$ to $z=1+2i$. $x9t) = t; y(t) = 2t$ so $\frac {dx}{dt} = 1 $ and $\frac{dy}{dt} = 2$.
3-25 SOS Problem 8.80. Find a bilinear transformation that maps the circle $|z – 1|= 2$ onto the line $ x + y =1$. Solution: Step 1. Understanding the problem: For a bilinear transformation $w=T(z)$ : circle $\to$ circle or line.
A bilinear transformation is determined by the images of 3 points. Step 2: Plan: Take 3 points on circle and assign them to 3 points
on line : use that to determine a bilinear transformation that
accomplishes the desired result. Step 3: Execute plan: 3 points on the circle: $z_1= 3, z_2=-1, z_3 = 1+2i. w_1=1 , w_2=i, w_3=2-i$.
Side work to find transformation:
Method A. Since this transformation must involve inversion to take a
circle to a line, Solve simultaneous equations for $T$ with $\delta=1$. Method B. Use the invariance of the cross ratio: $\frac
{(w-w_1)(w_2-w_3)}{(w-w_3)(w_2-w_1)}= \frac
{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} \\
\frac {(w-1)(i-(2-i))}{(w-(2-i))(i-1)}= \frac {(z-3)(-1-(1+2i))}{(z-(1+2i))(-1-3)} \\
\frac{(w-1)2}{(w-2+i))}=\frac {(z-3)(1+i))}{(z-1-2i))(2)}$
New equation from cross multiplying: $4(w-1)( z-1-2i)=(z-3)(1+i) (w-2+i) $
Collect w terms on one side of equation: $ w(4z-4-8i)-w(z(1+i)-3(1+i))= 4(z-1-2i) +(z-3)(1+i)(-2+i)$
$w*[(4z-4-8i)-z(1+i)+3(1+i)]= 4(z-1-2i) +(z-3)(1+i)(-2+i)$
So $w = \frac{4(z-1-2i) +(z-3)(1+i)(-2+i)}{(4z-4-8i)-z(1+i)+3(1+i)}$. Simplify to obtain the desired transformation. 3-27 [Draft] Part of these notes are taken from Module for The
Cauchy-Goursat Theorem
A closed curve has the property γ(a)=γ(b) and a simple curve has the
property $\gamma(t) \ne \gamma(s)$; for $ s \ne t$ except when
$\{s,t\}=\{a,b\}$.
Jordan Curve Theorem: For a closed, simple curve, a point not on the curve is either inside or outside the curve.
Let the simple closed contour C
have the parametrization $C: z(t) = x(t) + y(t) i$ for $a \le t \le b$. Recall
that if $C$ is parametrized so that the
interior of $C$ is kept on the left as
4z(t)$ moves
around $C$, then we say that
$C$ is oriented positively
(counterclockwise); otherwise, $C$ is
oriented negatively (clockwise).
If $C$
is positively oriented, then $-C$ is
negatively oriented. The figure below illustrates the concept
of positive and negative orientation.
Fact: If the point is inside the curve and you draw a line from the point to what is clearly outside the curve,
then the line will cross the curve an odd number of times. If the point is outside the curve and
you draw a line from the point to what is clearly outside the curve, then the line will cross the
curve an even number of times.
If you have an annulus and want to integrate over the boundary
curve, orient the inside circle in the direction opposite of
the outside circle. 3-30 More to come: Key ideas for the Cauchy Goursat Theorem.
A
domain D is said to be a simply
connected domain if the interior of any simple closed contour
C contained in D
is contained in D. In
other words, there are no "holes" in a simply connected
domain. A domain that is not simply connected is said to
be a multiply connected domain. Figure 6.16 illustrates
uses of the terms simply connected and multiply connected.
4-1 Review of materials on Quiz 3. 4-3 Completion of proof of Cauchy's Theorem for a triangle. [Start of Applications of CT].
See SOS for fundamental outline of argument.
We have found a nested sequence of similar triangles $\Delta_k$
with $|\int_{\Delta}f(z) dz| \lt 4^n|\int_{\Delta_n}f(z) dz|$.
We also have located a number $z_0$ where $z_0 \in \Delta_k$ for all $k$.
Since $f$ is analytic on a domain that contains $z_0$, $\lim_{z \to z_0}\frac {f(z)-f(z_0)}{z-z_0} = f'(z_0)$.
Rephrasing this in terms of $\epsilon's$ and $\delta's$, we have that:
Given $\epsilon \gt 0$ there is a $\delta \gt 0$ so that
$\frac {f(z)-f(z_0)}{z-z_0} - f'(z_0) =\eta \lt \epsilon$ whenever $|z-z_0| \lt \delta$.
Thus for $n$ large enough, $f(z)= f(z_0) + f'(z_0)(z-z_0) + \eta (z-z_0)$, so
$\int_{\Delta_n}f(z) dz = \int_{\Delta_n}f(z_0)dz + \int_{\Delta_n}f'(z_0)(z-z_0)dz + \int_{\Delta_n}\eta (z-z_0) dz$.
But $\int_{\Delta_n}f(z_0)dz = 0$ and $ \int_{\Delta_n}f'(z_0)(z-z_0)dz =0$.
So our attention turns to
$\int_{\Delta_n}\eta (z-z_0) dz$
Turning to the comparison of the modulus of the integral property we have
$|\int_{\Delta_n}\eta (z-z_0) dz| \le l(\Delta_n) * M$ where $M= max(\eta |z-z_0|)$ on $\Delta_n$.
But on $\Delta_n$, $|z-z_0| \lt \frac {l(\Delta) }{2^n}$,
$l(\Delta_n) \lt \frac {l(\Delta) }{2^n}$, and $\eta \lt
\epsilon$.
So we conclude that for $n$ large enough,
$|\int_{\Delta}f(z) dz| \lt 4^n|\int_{\Delta_n}f(z) dz| \lt 4^n
|\int_{\Delta_n}\eta (z-z_0) dz| \le 4^n*l(\Delta_n) * M \lt 4^n
\epsilon \frac {l(\Delta) }{2^n}\frac {l(\Delta) }{2^n} \lt \epsilon
l(\delta)^2 $.
Thus $|\int_{\Delta}f(z) dz| $is smaller than any positive real number, so $\int_{\Delta}f(z) dz =0.$
Week 11 Notes - still in draft form- more details may appear in the future.
Monday April 6:
Monday April 6th, 2015
Today, we initiated lecture by going over Cauchy's Integral Formula
(5.1) as stated on page 144 of our Schaum's Outlines Complex Variables
text.
A slight manipulation of (5.1) yields:
$\oint\limits_C\frac{f(z)}{z-a}dz=f(a)2\pi i$
which we became familiar with by looking at simple applications such as:
$(1)\oint\limits_{|z|=1}\frac{1}{z}dz=f(0)2\pi i=2\pi i$
where $f(z)=1$ for all $z$ and $a = 0$.
Similarly:
(2)$\oint\limits_{|z-i|=1}\frac{z^{2}+5z+3}{z-i}dz=2\pi i$
(3)$\oint\limits_{|z-1|=1}\frac{z^{2}+5z+3}{z-i}dz=\pi (2+5i)$
(4)$\oint\limits_{|z|=2}\frac{z^{2}+5z+3}{z^{2}+1}dz=
\oint\limits_{|z|=2}\frac{z^{2}+5z+3}{(z+i)(z-i)}dz$
$=\oint\limits_{|z-i|=1}\frac{z^{2}+5z=3}{(z+i)(z-i)}dz+
\oint\limits_{|z+i|=1}\frac{z^{2}+5z=3}{(z+i)(z-i)}dz$
=$\pi(2+5i)-\pi(2-5i)=10\pi i$
At this point we made the observation that finding complex integrals is easier than finding real integrals.
Section 5.2 of our Schaum's Outline's Complex Variables text lays out
all of the important consequences of Cauchy's Integral Formulas, one of
which includes The Fundamental Theorem of Algebra, which we get into
later in the week.
Wednesday April 8th, 2015
This week we have been working on complex integration using Cauchy's Integral Formulas.
We began lecture by looking at exercise 5.32 b) in our Complex Variable text. The problem statement is given as follows:
(1) Evaluate $\oint\limits_C\frac{e^{3z}}{z-\pi i}dz$ if $C$ is the ellipse $|z-2|+|z+2|=6$.
Our immediate goal was to verify whether $\pi i$ was on our ellipse or
not. A quick check using sample point $3i$ reveals that $\pi i$ is
outside of our ellipse since $3i$ is outside of our ellipse. Thus by
Cauchy's Theorem, $\oint\limits_C\frac{e^{3z}}{z-\pi i}dz=0$
since $\frac{e^{3z}}{z-\pi i}$ is analytic on our curve $C$.
We also looked at exercise 5.35 which asks to:
(2) Evaluate $\oint\limits_C\frac{e^{iz}}{z^{3}}dz$ where $C$ is the circle $|z|=2$.
A quick use of Cauchy's Integral Formula (5.2) with $a=0$ and $n=2$ will give us our solution (as the reader should verify).
Similarly,
(3)$\oint\limits_{|z|=1}\frac{cos(z)}{z^{3}}dz=\frac{2\pi i}{2i}f''(0)=-\pi i$ using (5.2).
We ended the lecture by looking at the Fundamental Theorem of Algebra
and how the factor theorem is a consequence of the division algorithm.
In the context of real numbers , polynomials with real coefficients
factor into a product of linear factors and irreducible quadratic
factors.
Friday April 10th, 2015.
Today we looked viewed a video of a 9 year old boy prove The Fundamental
Theorem of Algebra in various ways. We attempted to catch flaws in his
reasoning, but failed. See the video at: https://www.youtube.com/watch?v=SnUnkr3shDg
04-13: Sequences and Series
In the complex plane sequences are not tied down to the real line and have much more freedom.
Given
$\epsilon > 0$ if there exists $N$ such that if $k>N$, then $|z_k - L|
< \epsilon$, then we say $\lim _{n \to \infty} z_n = L$.
$(-1)^n$ alternates between $0$ and $1$, hence diverges.
Same is true about $i^n$ , but alternates between $1 , i , -1 ,$ and $- i$.
$(1/2)^n : 1 , 1/2, 1/4, 1/8, ... \to 0$.
In the complex plane $(1/2i)^n : 1, 1/2i, -1/4, -1/8i, ....$ but still goes to $0$.
If there exists a limit, then the limit is unique.
In the real numbers, $r^n \to 0$ if $-1 < r < 1$, $\to $ if $r=1$, and diverges if $r \le -1$ or $r>1$.
In the complex numbers, $z^n \to 0$ if $|z|<1$, $\to 1$ if $z=1$, and diverges if $|z| \ge1, z \ne 1$.
04-15:
If $|z|=1$ and $Arg(z)$ is a rational multiple of $\pi$
then the sequence $z^n$ cycles as it goes around and around the unit
circle.
We looked at sequences in the complex plane using GeoGebra.
If $|z|=1$ and $Arg(z)$ is an irrational multiple of $\pi$ the
sequence $z^n$ gets close to whatever we want (dense) on the unit
circle.
If $|z|>1$, then the sequence $z^n$ is getting farther away from any point.
Geometric series in complex plane discussed briefly.
04-17:
Looked at various series on GeoGebra.
Every bounded sequence of complex numbers has a convergent subsequence (a consequence of compactness).
The
ratio test works in the complex plane for complex series. The series
convergence will be absolute convergence- which inplies convergence of
the series.
The
Taylor series: $\sum_{n=0}^{\infty} f ^{n}(0) \frac{z^n}{n!}= f(x)$ ,
Theorem: Whenever the Taylor series converges, it will converge to the
original function.
4-20
Looking at $f(z) = \frac {e^z}{z^5}$, we have a pole of order 5 at the origin, because $z^5 f(z)$ is analytic at $z=0$.
$f(z) = e^{1/z}$ has an isolated essential singularity at the origin
becauseany power of z, $z^n f(z)$, still has a singularity at $z=0$.
$f(z)=\frac{\sin(z)}z$ has a removable singularity. (just like on the real number function $f(x)=\frac{\sin(x)}x$).
$ f(z) = \frac1{1-z}$ has a simple pole when $z=1$.
Taylor's Theorem (for polynomial approximations) was proved, can be seen in
our textbook.
We noted that the remainder of $f(z) -T_n(z)$ can be calculated by
bounding the largest magnitude of $w = f(z)$ on the circle times the circumference of the circle.
We
also noted that the Cauchy Integral Formula doesn’t apply to the real numbers,
only complex numbers.
Problem 6.37 was looked at . The first term is $\frac 12$ and the
ratio between successive terms was noted to be $\frac z2$ , so using the geometric series
formula, we calculated that the series converges to $\frac1{2-z}$.
Factoring out a $\frac 1z$ from the entire series, we have that the
series converges if the magnitude $|2z| \lt 1$. Solving for $z$, we have that
the magnitude has to be less than $\frac 12$, and so the series converges to the
product $\frac1z \cdot \frac 1{2z-1}$.
4-22
We started the day with showing that the $\tan(z)$ can be
expressed as a set of taylor polynomials for $|z| \lt \frac{\pi}2$.
The trick to solve this problem is to take multiple
derivatives, and using Taylors theorem to expand this to order 7. Wolfram alpha
can be used to check the work.
When looking at $f(z)= e^z$ and at its expansion about $z=1$, we noted that
plugging $1$ for $z$ gave us the
polynomial $e+e(z-1)+e\frac{(z-1)^2}{2!}+...+e\frac{(z-1)^n}{n!} + ...$ And so on and so on.
The next example we looked at was $f(z)= \frac {e^z} {z^3}$. This function
has a pole of order 3 at the origin for obvious reasons. Now in order to relate
this function to an analytic function, we multiply it by $z^3$ which gets rid of the problem at the origin
and makes it nice.
We first write $f(z)$ out as a Laurent Series by factoring out a
$\frac 1{z^3}$ from $e^z$. This gives the Laurent series:
$f(z)= \frac 1{z^3} (1 + z + \frac {z^2}{2!} + ...+\frac{z^n}{n!} + ... =
\frac 1{z^3} + \frac 1{z^2} + \frac 1{z 2!} +
...+\frac{z^{n-3}}{n!} + ... $
Looking at example 6.91, we are asked to give $\frac1{z-3}$ in a Laurent
Series Valid for $|z|<3$.
We first factor out a $\frac{-1}3$ from the function to give us a
form we can work with. Letting $u =\frac z3$, we have the function $f(z) = \frac {-1/3}{1-u}$.
Writing out the series, we show that it converges for values of magnitudes $|z| \lt 3$.
For magnitudes $|z| \gt 3$, turns out, you flip all
the
terms, $\frac1{z-3} = \frac1{1-3/z}$, then expand with geometric series
which will converge when $|\frac 3z| \lt 1$ or $|z| \gt 3$ and the
truth holds.
Problem 6.92, we want to expand $\frac z{(z-1)(2-z)}$. Using partial
fractions , we found $A=1$ and $B=2$, so$\frac z{(z-1)(2-z)} =
\frac 1{z-1} + \frac 2{2-z}$
.
(a) for $|z| \lt 1$: Writing each term as an its own series, we get that
the series is the sum from $\sum_0^{\infty}(-1 +\frac1{2^n})z^n$.
(b) For part two of the problem, $ 1 \lt |z| \lt 2$, the B term is
good, but the first part must be made into a Laurent Series, and therefore
"flipped" as in 6.91.
4-24
We talked about uniform continuity.
In general, uniform continuous implies continuous, but the converse usually doesn’t hold.
What it means for a function
to be uniformly continuous is that for every $\epsilon
>0$ there exists a $\delta >0 $ such that $|z-c|< \delta$
implies $|f(z)-a|<\epsilon$ for any $c$ in the domain of $f$. Meaning
that we
just can use one delta for the whole domain of the function with the
limiting values of the function falling
uniformly within the given epsilon bound.
We discussed a Theorem; If f is restricted to a
compact set, then
f being continuous implies uniform continuity, So for a bounded domain, a
continous functon is uniformly continuous for any closed subset. And so
COMPACTNESS => {Continuity =>UNIFORM CONTINUITY}.
A key example from real variable functions is $f(x) = \frac 1x$ where
the function is continuous for $(0,1)$, but not uniformly continuous. When
$a$ is close to $0$ the choice of $\delta$ requires them to be smaller
because the magnitudes in $\frac 1x$ becomes larger.
Consider the function $\frac1z$. This function is has an isolated singularity at
the origin, so it can’t be uniform continuous for a domain containing $0$.
Consider the same function but with the
restricted domain of $|z| \ge 1$. Because now we can restrict how big our y
values get, we get uniform continuity on this domiain.
4-27 To Appear
4-29: More to come.
$C_2: \gamma(t)=Re^{it}$ with $0 \le t \le \pi$.
Outline for showing $\lim_{R\to \infty}\int_{C_2} \frac 1{1+z^2} dz =0$.
Assume $R \gt 1$.
$|\int_{C_2} \frac 1{1+z^2} dz| = |\int_0^\pi \frac 1{1+R^2e^{2ti}}
Rie^{it}dt| \le \pi \ max\{ |\frac 1{1+R^2e^{2ti}}| \cdot R\}$.
Note that on $C_2$, $|1-R^2| \le |1+R^2e^{2ti}| $ so $|\frac R {1-R^2}| \gt |\frac 1{1+R^2e^{2ti}}| \cdot R\}$.
So $|\int_{C_2} \frac 1{1+z^2} dz| \le \pi | \frac R {1-R^2}|$ .
As $R \to
\infty$, we have $\frac R {1-R^2}\to 0$, so we conclude $\lim_{R\to \infty}\int_{C_2} \frac 1{1+z^2} dz =0$.