Thank you for your input.
I totally forgot about complex analysis. I have also taken that class using Palka's Intro to Complex Function Theory. At my university, these are the "most advanced" undergraduate courses.
The algebra class was two-semesters long but as you have already noted...
My curriculum was heavy proofs from the very beginning.
My honors analysis class used Spivak. I also took a masters level analysis class that used baby Rudin. My abstract algebra class used Dummit and Foote. My other classes which include linear algebra, differential equations, number theory...
Its been a long ride. 4 years ago when I started college, I started as a finance major. I excelled in all of my classes but found the material to be a little boring so I changed to economics. I continued to get stellar grades and even now have nothing but A's in all of my economics courses. I...
I figured it out for future reference to anybody.
Use ##sin(z)## to take the infinite strip to ##\mathbb{C}\sim\{w:|\Re(w)|\geq 1## and ##\Im(w)=0\}##. Then rotate this by multiplying by ##i## and finally use ##Arctan(w)## to take it back to the infinite strip.
Homework Statement
Find a conformal mapping of the strip ##D=\{z:|\Re(z)|<\frac{\pi}{2}\}## onto itself that transforms the real interval ##(-\frac{\pi}{2},\frac{\pi}{2})## to the full imaginary axis.
The Attempt at a Solution
I tried to map the strip to a unit circle and then map it back to...
I know that for the tangent unit vector ##t##, normal unit vector ##n##, and binormal unit vector ##b## that ##b=t\times n## and ##n=b\times t##. Is it true that ##t=n\times b##?
**Edit** Ah! Yes it is. Nevermind. I should have known this was true.
This problem comes from the first chapter in the textbook which is an introduction complex analysis. Picard's theorem comes in chapter 4.
Do you know if there is any way to parametrize S without ##x=|z|\cos\theta##, ##y=|z|\sin\theta##?
I can't seem to picture it. In class I was shown that if z=x and Im(z)=0 then e^z was a circle and if Re(z)=0 and Im(z)=y then e^y was a vector that pointed outwards from the origin at an angle of y. Combining these together all I can see is two circles, one inside the other bounding the area...
If ##f(z)=\frac{1}{z}##, ##f(S)## would be a disk with a hole inside it centered at the origin with radius ##\frac{1}{r}## but in this case, the function is the exponential.
I think ##f(z)=e^z## maps z to a circle on the complex plane of radius Re(z) so I'm tempted to say f(S) is a mess of...
If ##0<|z|<r## then we have ##\frac{1}{r}<\frac{1}{|z|}## and ##\frac{1}{|z|}\rightarrow\infty## as ##|z|\rightarrow 0## but ##0<|z|## so we can safely say ##\frac{1}{|z|}<\infty##.
Homework Statement
Determine ##f(S)## where ##f(z)=e^{\frac{1}{z}}## and ##S=\{z:0<|z|<r\}##.
*Edit: The function f is defined as ##f:\mathbb{C}\rightarrow\mathbb{C}##.
The Attempt at a Solution
I am a little confused as to what this problem is asking me to do. What I did was:
Let...
Ah! Then since ##z_{0}## is contained in ##f(\mathbb{C})##, this is a contradiction because ##G## does not contain its boundary. If it did then ##G## would also have to be closed and the only sets which are both open and closed in ##\mathbb{C}## are ##\emptyset## and ##\mathbb{C}## but since...
By Bolzano-Weierstrauss, there exists a convergent subsequence of ##z_{n}##,##z_{n_{k}}## which converges to some ##z\in\mathbb{C}##. Then since ##f## is continuous, ##f(z_{n_{k}})\rightarrow f(z)=z_{0}##.
I am not sure how to proceed. I keep thinking that the goal is to derive a contradiction...
Homework Statement
Let a continuous function ##f:\mathbb{C}\rightarrow\mathbb{C}## satisfy ##|f(\mathbb{C})|\rightarrow\infty## as ##|z|\rightarrow\infty## and let ##f(\mathbb{C})## be an open set. Then ##f(\mathbb{C})=\mathbb{C}##.
The Attempt at a Solution
Suppose for contradiction that...
Homework Statement
Let ##f## be a ##2\pi## periodic function. Let ##\hat{f}(n)## be the Fourier coefficient of ##f## defined by
$$
\hat{f}(n)=\frac{1}{2\pi}\int_{a}^{b}f(x)e^{-inx}dx.
$$
for ##n\in\mathbb{N}##. If ##\overline{\hat{f}(n)}=\hat{f}(-n)## show that ##f## is real valued.
The...
Ah! Thanks! I see what I was doing wrong now. I was plugging in ##y(x)=z(x)## to try to get the second solution instead of the correct ##y(x)=cx^{2}\ln x+z(x)##. Thanks for your help!
Homework Statement
I want to find two linearly independent solutions of
$$
x^{2}y''-2x^{2}y'+(4x-2)y=0.
$$
The Attempt at a Solution
The roots to the indicial polynomial are ##r_{1}=2## and ##r_{2}=-1##.
I found one solution which was ##x^{2}## and I am having trouble finding the...
Hmm. The way the proof goes in my text for ##\zeta(s)=\prod_{p}\left(\frac{1}{1-p^{-s}}\right)## is that a finite product, ##P_{k}(s)## is defined. The author then uses the fact that the finite product is equal to a finite sum with the general term ##\frac{1}{n^{s}}##. The author then goes to...
Homework Statement
I wish to prove that for s>1
$$
\sum\limits_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}.
$$
The Attempt at a Solution
(1) I first showed that
$$
\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}.
$$
It was a given theorem in the text that
$$...
Homework Statement
Suppose ##\phi(x)## is a function with a continuous derivative on ##0\leq x<\infty## such that ##\phi'(x)+2\phi(x)\leq 1## for all such ##x## and ##\phi(0)=0##. Show that ##\phi(x)<\frac{1}{2}## for ##x\geq 0##.
The Attempt at a Solution
I tried to solve this like I...
I am currently taking a mathematics course where the exams are three parts. The first part is just to regurgitate formulations of theorems and definitions. The second part is to prove important theorems that were presented in class and in the assigned readings. The third part is what all of my...
In case anyone else ever has this problem, I was able to figure out why the sets are equivalent. By a previous theorem about 100 pages or so before this lemma, Hardy proved that if (g,p)=1 and 1,2,...,p-1 are a set of incongruent residues mod p then g,2g,...,g(p-1) is also such a set.
I would...
I am reading Hardy's Intro to the Theory of Numbers and I am currently trying to work through the proof of Von Staudt's Theorem. Hardy first proves the following lemma.
$$
\sum\limits_{1}^{p-1}m^{k}\equiv -\epsilon_{k}(p) (\mod p).
$$
Proof: If ##(p-1)|k## then ##m^{k}\equiv 1## by Fermat's...
I am trying to find the Jordan Decomposition of T. That is what it is called in my book. From a quick search it doesn't look like it goes by the name of Jordan Decomposition on wikipedia.
In my book D is defined to be a diagonalizable linear transformation while N is defined to be a...
Hmm. I am choosing the new basis ##w_{1}=v_{1}-iv_{2}## and ##w_{2}=v_{1}+iv_{2}## which gives me
$$
B=\begin{pmatrix}
0 & 0 \\
0 & 2 \\
\end{pmatrix}
$$
and
$$
S=\begin{pmatrix}
1 & 1 \\
-i & i \\
\end{pmatrix}
$$
which gives me the SB=AS that I need but then when I do the Jordan...
Homework Statement
I have a linear transformation T defined by
$$
T(v_{1})=v_{1}+iv_{2}\\
T(v_{2})=-iv_{1}+v_{2}\\
$$
and I want to find a triangular matrix B of T and an invertible matrix S such that SB=AS where A is the matrix of T with respect to the basis ##\{v_{1},v_{2}\}##.
The Attempt...
I am a little confused on how this shows that any v can be expressed as a sum of v1 and v2. It looks to me like it just shows that any vector is either in v1 or v2.
Homework Statement
Let ##T\in L(V,V)## such that ##T^{2}=1##. Show that ##V=V_{+}\oplus V_{-}## where ##V_{+}=\{v\in V:T(v)=v\}## and ##V_{-}=\{v\in V:T(v)=-v\}##.
The Attempt at a Solution
I was given a theorem that said that ##V## is the direct sum if and only if every vector in ##V## can...
This was very helpful. I will definitely work on your suggested exercises in 3. As for the binomial coefficient, I will find it in my notes and get back to you.
Thats right. There should be a bar over d to signify its the upper density. Then d(A) represents the natural density is what my professor said today.
Sorry about that. I can't seem to edit my original post anymore.
I am sorry for not being more clear. ##|A \bigcap {1,2,...n}|## stands for the number of elements in the intersection of ##A## and ##{1,2,...,n}## as ##n\rightarrow\infty##.
Ahh. I am talking about things like shift invariance of density. For example, if we define
$$
d(A)=\lim\limits_{N\rightarrow\infty}\sup\frac{|A\cap\{1,...,N\}|}{N}
$$
then
##d(A-t)=d(A)## for all ##t\in\mathbb{Z}##.
In general, the problems usually consist of finding out whether a given...
I am currently taking a number theory course. The professor who teaches the course is very well known in his research field which is Ergodic Theory.
I find that he really likes to assign problems that deal with the density of a certain set. Sometimes the problems would be about proving...
I am currently using this text for one of my classes. I find it difficult to read from this book and I often have to use other materials to supplement the topics in this book. Even the newest edition is riddled with errors. Its as if none of the editors ever actually read the book before...
Homework Statement
Let x be any positive real number and suppose that ##x^2-ax-b=0## where ##a,b## are positive. I would like to use the equation that I provided in relevant equations which I proved to prove that
$$
\sqrt{\alpha^{2}+\beta}=\alpha+\cfrac{\beta}{2 \alpha+\cfrac{\beta}{2...
I tried to expand this further into
$$
(v_{1}-\lambda_{1}v_{1})^{2}+\cdots+(v_{n}-\lambda_{n}v_{n})-(v_{1}-(v,v_{1})v_{1})^{2}-\cdots-(v_{n}-(v,v_{n})v_{n})^{2}
$$
which is equal to
$$...
I can see how the expression
$$
\left|\left|v-\sum\limits_{i=1}^{n}\lambda_{i}v_{i}\right|\right|^2-\left|\left|v-\sum\limits_{i=1}^{n}(v,v_{i})v_{i}\right|\right|^2.
$$
becomes the difference between two inner products:
$$
(v-x,v-x)-(v',v')
$$
and writing that out I do get a sum of squares...
Homework Statement
Let ##v_{1},...,v_{m}## be an orthonormal set of vectors in ##V##. Let ##v\not\in S(v_{1},...,v_{m})##. Show that the vector ##v'=v-\sum\limits_{i=1}^{n}(v,v_{i})v_{i}## given by the Gram-Schmidt process has the shortest length among all vectors of the form ##v-x## for ##x\in...