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Jan
24
comment Prove the uniformity of the Cantor/Lebesgue function defined on $A^c$ where $A$ is a Cantor set on $[0,1]$
Thank you for a lucid explanation. Is this a stylistic choice on the author's part that is reasonable to expect most readers to understand or would most writers make the distinction between $A$ and $A_k$?
Jan
7
comment Why do we usually construct the Lebesgue Measure on finite outer measure sets before arbitrary measurable sets?
I'm still a little confused as to why that's an issue. Can you elaborate?
Jan
7
comment Why do we usually construct the Lebesgue Measure on finite outer measure sets before arbitrary measurable sets?
Surely the first method does not imply that every set had measure $\infty$, only those that have inner measure $\infty$ have measure $\infty$, which is what we want, correct?
Dec
24
comment Prove that every nonempty open subset $G$ of $\mathbb{R}^n$ can be expressed as a countable union of nonoverlapping closed rectangles.
@bof, I suppose the problem could be wrong, but I am quite sure that I am quoting the problem correctly. It is from the book Lebesgue Integration on Euclidean Space by Frank Jones. It is Chapter 2 Section A Problem 9.
Dec
23
comment Prove that every nonempty open subset $G$ of $\mathbb{R}^n$ can be expressed as a countable union of nonoverlapping closed rectangles.
Why does the density of $\{x_n\}$ mean that all points in $G$ must have been covered? Aren't the points in $G$ uncountable? Maybe I have some trouble with the terminology "countably dense" which I have not encountered before.
Dec
23
comment Prove that every nonempty open subset $G$ of $\mathbb{R}^n$ can be expressed as a countable union of nonoverlapping closed rectangles.
For this questions, the rectangles must not overlap and they are $n$-dimensional. They could also be cubes.
Dec
16
comment Calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ using complex variables
Lightbulb moment. Of course. Thank you!
Dec
16
comment Calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ using complex variables
How do you get from $\left(e^{-i8\pi/5} - e^{-i6\pi/5} + e^{-i16\pi/5} - e^{-i12\pi/5}\right)$ to $\left(e^{i2\pi/5} - e^{i4\pi/5} + e^{i4\pi/5} - e^{-i2\pi/5}\right)$?
Dec
15
comment What conditions are necessary on $a,b,c,d$ so that the Mobius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point?
@DanielFischer, on your last question, no. I am only assuming that $ad-bc \neq 0$. I thought the "standard" definition of a Mobius transform implied that $a,b,c,d \in \mathbb{R}$ -- I guess I'm mistaken on this point. Also, I would be happy to upvote your comments if you were interested in submitting them as an answer (since you are addressing my question directly).
Dec
15
comment What conditions are necessary on $a,b,c,d$ so that the Mobius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point?
@DanielFischer The reason I believe it should lie in $\mathbb{R}$ is because $a,b,c,d \in \mathbb{R}$. If there is only one fixed point, then (if I'm right) the discriminant is zero and what remains is in $\mathbb{R}$. I think the only way for a complex number to "arise" is if the discriminant < 0. I could be mistaken.
Dec
14
comment Compute $\int_0^\infty \frac{dx}{x^5+1}$ using a contour in the upper half complex plane that encloses one of the roots of $z^5+1=0$
Thanks at least for confirming my suspicion that the hint was incorrect.
Dec
14
comment Describe the Riemann surface for $w=z^2-1$.
thank you for a lucid explanation!
Dec
12
comment Describe the Riemann surface for $w=z^2-1$.
Can you explain or go into some detail as to why this choice of cut makes sense? I am having a difficult time visualizing how the cut would work..
Dec
12
comment Describe the Riemann surface for $w=z^2-1$.
Does the multivalued nature of the function, and hence the necessity of the riemann surface, follow from the fact that the inverse of the function is multivalued?
Dec
8
comment Show the level curves of $\log|f(z)|$ are orthogonal to those of $\operatorname{arg}(f(z))$.
If we write $f(z) = re^{i\theta}$, then we know that the $\log f(z)$ depends on the size of $r$ and it's angle is defined by $\theta$. Perhaps I'm on the wrong track, I'm not sure what that helps achieve.
Dec
7
comment For what values $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge.
What technique have you used to split the integral after the first inequality?
Dec
7
comment For what values $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge.
Great approach! Let me see what I can do...
Dec
6
comment Without using the beta function, find values $q,r$ such that the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?
This is what I was looking for. I understand your argument but, to clarify, how do we know that the integral cannot go to infinity anywhere else besides the endpoints?
Dec
2
comment $\{f_n\} \in \mathcal{R}[a,b]$, converges pointwise to $f\in \mathcal{R}[a,b]$. Prove $\lim_{n\to\infty}\int_a^b f_n \, dx = \int_a^b f \, dx$.
That second article is fascinating! Thank you!
Nov
28
comment If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$
Ok. There is a follow up question which says "Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{x \in \mathbb{R} |f(x) = 0\}$ is a closed subset of $\mathbb{R}$.". But based on what you are saying, this question doesn't make any sense. Can you help in my understanding?