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1. ### Schools Planning for graduate school in mathematics

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...
2. ### Schools Planning for graduate school in mathematics

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...
3. ### Schools Planning for graduate school in mathematics

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...
4. ### Conformal mapping of an infinite strip onto itself

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.
5. ### Conformal mapping of an infinite strip onto itself

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...
6. ### Confirmation of cross products

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.
7. ### Finding f(S) for z = e^(1/z)

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##?
8. ### Finding f(S) for z = e^(1/z)

I'm confused. I don't see the point in this exercise if the work I was doing before your input was correct.
9. ### Finding f(S) for z = e^(1/z)

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...
10. ### Finding f(S) for z = e^(1/z)

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...
11. ### Finding f(S) for z = e^(1/z)

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##.
12. ### Finding f(S) for z = e^(1/z)

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...
13. ### Complex Analysis problem

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...
14. ### Complex Analysis problem

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...
15. ### Complex Analysis problem

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...
16. ### Fourier coefficients

Woops! This was an if and only if problem and I was having trouble with the converse part. Sorry for the confusion.
17. ### Fourier coefficients

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...
18. ### Using Frobenius Method

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!
19. ### Using Frobenius Method

The equation is ##x^{2}y''-2x^{2}y'+(4x-2)y=0##. Unless I'm completely crazy, ##y(x)=x^{2}## is a solution.
20. ### Using Frobenius Method

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...
21. ### Inverse of the Riemann Zeta Function

Yea sorry about that. When I realized my mistake, it was already too late to change it.
22. ### Inverse of the Riemann Zeta Function

Alright. Thanks!
23. ### Inverse of the Riemann Zeta Function

Yea. Maybe it's because I don't deal with infinite products often and need more experience with them.
24. ### Inverse of the Riemann Zeta Function

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...

47. ### Show that the Gram-Schmidt process gives the shortest length

Ahh! Yes. That does it. I got too bogged down with my notation before. Thanks.

50. ### Show that the Gram-Schmidt process gives the shortest length

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...