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  1. N

    Complex Analysis: Open Mapping Theorem, Argument Principle

    OK thanks, just to be clear, are you saying I did parts (a) and (b) correct? EDIT: Actually, I clean my answer part (a) up a little bit below. It's not a significant change but it should make my reasoning more clear. --------------- This is true. By hypothesis, there exists ##z_n\in...
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    Complex Analysis: Open Mapping Theorem, Argument Principle

    Homework Statement In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample. (a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...
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    Complex Analysis: Series Convergence

    Sorry, wasn't trying to be rude. Your help was much appreciated and during the process I learned a lot. However, this solution is from the professor and was revealed before I was able to finish the problem using your strategy. For the sake of completeness I thought I should post it. Yes, the...
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    Complex Analysis: Series Convergence

    Here's an outline to a solution: ##f## and ##g## holomorphic implies ##\frac{f}{g}(z)=\sum_{n=0}^{\infty}c_n(z-a)^n## is holomorphic for ##|z-a|<r## By the product rule for power series ##f(z)=\frac{f}{g}(z)g(z)=\sum_{n=0}^{\infty}[\sum_{j=0}^{n}c_{j}b_{n-j}](z-a)^n## Then by uniqueness of the...
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    Complex Analysis: Series Convergence

    This is probably obvious but just to be clear does ##f_{2N}=\sum_{n=0}^{2N}a_nz^n=a_0+a_1z^n+\cdots + a_{2N-1}z^{2N-1}+a_{2N}z^{2N}=f_N+\sum_{n=N+1}^{2N}a_nz^n## ? Also, I'm not 100% clear on what you mean by a "grid". However, I wrote out each term of ##\sum_{j=0}^{p}c_jb_{p-j}## thinking it...
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    Complex Analysis: Series Convergence

    ##q_Ng_N=\sum_{n=0}^N c_n(z)^n \sum_{n=0}^N b_n(z)^n## ##=\bigg(\sum_{n=0}^N (a_n/b_0-\sum_{j=0}^{N-1}\frac{c_jb_{n-j}}{b_0})(z)^n\bigg) \sum_{n=0}^N b_n(z)^n## ##=\bigg[\sum_{n=0}^N \frac{a_n}{b_0}(z)^n-\sum_{n=0}^N \bigg(\sum_{j=0}^{N-1}\frac{c_jb_{n-j}}{b_0}\bigg)(z)^n \bigg]\sum_{n=0}^N...
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    Complex Analysis: Series Convergence

    Note: I'm just saving my progress so far. I still need to work on the stuff below this note some more. Hope that's OK. ##q_Ng_N=\sum_{n=0}^N c_n(z)^n \sum_{n=0}^N b_n(z)^n## ##=\sum_{n=0}^N (a_n/b_0-\sum_{j=0}^{N-1}\frac{c_jb_{n-j}}{b_0})(z)^n \sum_{n=0}^N b_n(z)^n## ##=\sum_{n=0}^N...
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    Complex Analysis: Series Convergence

    I get ##f_N=q_Ng_N=## ##\implies \sum_{n=0}^N a_n(z-a)^n=\sum_{n=0}^N c_n(z-a)^n \sum_{n=0}^N b_n(z-a)^n## ##\implies \sum_{n=0}^N a_n(z-a)^n=\sum_{n=0}^N (a_n/b_0-\sum_{j=0}^{N-1}\frac{c_jb_{n-j}}{b_0})(z-a)^n \sum_{n=0}^N b_n(z-a)^n## ##\implies \sum_{n=0}^N a_n(z-a)^n=\sum_{n=0}^N...
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    Complex Analysis: Series Convergence

    Let ##q_N## be the partial sum of the formal quotient given in the problem and let ##f_N##, ##g_N## and ##r_N## be the partial sums of the taylor series for f,g and the remainder of their quotient respectively. Then I want to show $$\lim_{N\to\infty}(f_N=q_Ng_N+r_N)<\infty$$? Say I can show...
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    Complex Analysis: Series Convergence

    The limit of the partial sums converge?
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    Complex Analysis: Series Convergence

    Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...
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    Complex Analysis: Identity Theorem

    A zero sequence would be a more apt descriptor then. So, a zero sequence is a sequence \{z_k\} such that f(z_k)=0 for all k and where \lim_{k\to\infty} z_k\to a. In my case a=0.
  13. N

    Complex Analysis: Identity Theorem

    Homework Statement Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique? (a) f(1/n)=4 for n=1,2,\dots (b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots A side question: Is corollary 1 from my...
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    Complex Analysis: Special Power Series

    Sorry, I'm not sure why my latex commands aren't taking.
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    Complex Analysis: Special Power Series

    It's OK to delete this. Posted it on accident.
  16. N

    Complex Analysis: Special Power Series

    Homework Statement Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex]. Homework Equations R is the radius of convergence and the derived series is the term by term...
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    Complex Analysis: Research

    That's what I thought at first and I suppose the class will get to Cauchy-Riemann eventually. But the instructor stressed the fact that we were not working in the complex numbers for this problem. The Theorem he was looking for is from Advanced calculus. He probably wants to demonstrate the...
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    Complex Analysis: Research

    I found it thanks! It's kind of a long theorem but if you're interested to know what it is let me know and I'll type. It doesn't have a distinct name that I can just reference for you.
  19. N

    Complex Analysis: Research

    Homework Statement This isn't a standard homework problem. We were asked to do research and to find a theorem of the form: If something about the partial derivatives of u and v is true then the implication is ##D(u,v)## at ##(x_0,y_0)## exists from ##R^2## to ##R^2## Homework Equations The...
  20. N

    Real Analysis

    OK thanks, I guess saying ##d(O_n, O^c)\ge 1/4## is just another way of writing what the lower bound is. I was probably overthinking ( possibly under-thinking) things. Does part (d) seem correct?
  21. N

    Real Analysis

    The distance between the two sets is ##inf d(x_i, y_i)## where ##x_i\in O_n##, ##y_i\in O^c##. But the problem I see is that by part (a) ##O_n## is open (but bounded) hence the lower bound of ##d(O_n, O^c)## is not attainable even though ##O^c## is closed. Also, the limits don't coincide...
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    (Solved) Real Analysis: Hardy Littlewood

    I was stuck on the first step. I was able to work in reverse from the solution but felt like I was missing a key idea doing it that way. Namely, ##\sup_{x\in B} \frac{1}{m(B)} \int_B \frac{1}{|x|(ln\frac{1}{x})^2}\ge \frac{1}{2|x|}\int_{-|x|}^{|x|} \frac{1}{|x|(ln\frac{1}{x})^2}## from...
  23. N

    Real Analysis

    Homework Statement Let ##O## be a proper open subset of ##\mathbb{R}^d## (i.e.## O## is open, nonempty, and is not equal to ##\mathbb{R}^d##). For each ##n\in \mathbb{N}## let ##O_n=\big\{x\in O : d(x,O^c)>1/n\big\}## Prove that: (a) ##O_n## is open and ##O_n\subset O## for all ##n\in...
  24. N

    (Solved) Real Analysis: Hardy Littlewood

    Homework Statement Establish the Inequality ##f^*(x)\ge \frac{c}{|x|ln\frac{1}{x}}## for ##f(x)=\frac{1}{|x|(ln\frac{1}{x})^2}## if ##|x|\le 1/2## and 0 otherwise Homework Equations ##f^*(x)=\sup_{x\in B} \frac{1}{m(B)} \int_B|f(y)|dy \quad x\in \mathbb{R}^d## The Attempt at a Solution...
  25. N

    Lebesgue Integral: Practice Problem 2

    ##E_n\subset E_{n+1} \implies f_n ##is a monotone increasing sequence so ##f_n<f_{n+1}## from ##f_n=f\mathcal{X}_{E_n}## is is clear that ##f_n\to f## Unfortunately the test wasn't anywhere near as easy as the practice problems..lol
  26. N

    Lebesgue Integral: Practice Problem 2

    EDIT: Deleted my reply. The test was so hard that he decided to make it a take home exam at the last minute. I'll repost my response after the due date.
  27. N

    Lebesgue Integral: Practice Problem 2

    Homework Statement Let ##f:\mathbb{R}\to \mathbb{R}## be a nonnegative Lebesgue measurable function. Show that: ##lim_{n\to\infty}\int_{[-n,n]}f d\lambda=\int_{\mathbb{R}}f d\lambda## Homework Equations The Attempt at a Solution Let ##E_n=\{x:-n<x<n\}## then write ##f_n=f\mathcal{X}_{E_n}##...
  28. N

    Lebesgue Integral: Practice Problem

    Homework Statement Suppose that f is a nonnegative Lebesgue measurable function and E is a measurable set. Let A = {x ∈ E : f(x) = ∞}. Show that if ##\int_E f dλ < ∞## then ##λ(A) = 0## Homework Equations The Attempt at a Solution [/B] Let ##\phi(x)=\sum_{x\in A}a_i\mathcal{X}_{A_i}(x)## so...
  29. N

    Properties of Lebesgue Integral

    Ack! That's not good, this was the professors practice problem for the upcoming test. Also, I double checked and the problem is stated here as he gave it to the class. Please disregard my request for assistance.
  30. N

    Properties of Lebesgue Integral

    Here's a little more detail on my attempt ## \int (f-g)\le liminf_{n\to\infty}\int{(f_n-g_n)}## ##\implies \int{f}-\int{g}\le liminf_{n\to\infty}\int{f_n}-liminf_{n\to\infty}\int{g_n}## ##\implies -\int g -limsup_{n\to \infty}\int{-g_n} \le -\int f -limsup_{n\to \infty}\int{-f_n} ## ##\implies...
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