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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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.
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...
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?
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...
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...
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...
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...
##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
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.
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}##...
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...
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.
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...