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7h
comment Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
Tricky, indeed! This is a more explicit way of doing it than others have suggested, but the exposition is much appreciated.
7h
comment Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
You are using the product rule in the numerator to get to the last step, right? I think you might be missing a factor of $(1+x^2)$?
7h
comment Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
Thanks for your helpful answer. I think you may have made an error in the last step though, as I believe the limit equals -1, not 0.
7h
comment Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
I'm not familiar with the Big-O and little-o notation in how you're using it, though I appreciate that your answer is different from the others. Can you explain the notation so I can understand better?
7h
comment Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
I think you may have misread the question. The limit should be $-1$. Though your suggestion in the comments was helpful, I did need to apply l'hopital's again.
1d
comment Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise
This is very interesting. I'm not familiar with the sinc function. Is your argument invalid of we replace sinc with $\sin$?
2d
comment Sequence of $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$.
@copper.hat I was looking for verification of my attempt. I have modified my question and added a proof-verification tag. Sorry about that.
Nov
22
comment Statement of maximum modulus principle and question
I did not realize that was the implied definition of a domain. Is there a reason for why a connected open subset is defined as a domain and not a closed subset?
Nov
20
comment Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
Does the third follow from the fact that we want to map $1/2 \to m$ and $0 \to -m$? Also, are there any good resources for learning more about these? Book recommendations maybe? Based on your answer, it seems like there's much more to linear fractional transformations than my book describes.
Nov
20
comment Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
Thank you! How did you obtain the first two relations from my guess? You must have used my last equality somehow but I'm not sure how you manipulated it to achieve $a+b=c+d$ and $a-b=d-c$.
Nov
20
comment Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
@DanielFischer No, actually I am just learning about linear transformations. I'm awake that the interversion $w=1/z$ maps circles and lines to circles and lines. I am also aware of the form $w=(az+b)/(cz+d)$. Although, I'm not specifically aware of the automorphism of the unit disk.
Nov
20
comment Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
@DanielFischer, Would this mean that 1 maps to -1 and -1 maps to 1? Or that 1 maps to 1 and -1 maps to -1? If you have time for an answer or explanation, that would be great! Thanks.
Nov
20
comment What Mobius transformation maps the circles $|z-\frac{1}{4}| = \frac{1}{4}$ and $|z|=1$ onto two concentric circles centered at $w=0$?
What do you mean by pick "almost any values" for $a$ and $c$? Don't we need a third mapping to fully define the transformation?
Nov
20
comment For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?
@ThomasAndrews. I made an edit to my approach. Does this make sense?
Nov
20
comment For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?
I see I have some work to do. Is the general direction of my approach correct? As in, I tried to identify that the integral can only go to infinity near the endpoints, so we should find the appropriate values of $r,q$ there. How can I approach the problem in this way?
Nov
20
comment For what values $p,q$ does the improper integral $\int_0^1 x^p (1-x^2)^q dx$ converge?
Doesn't $(1-\sin(t)^2)^q = \cos^{2q}(t)$ rather than $(1-\sin(t)^2)^q = \cos^{2q}(t)\cos(t)$? What am I missing?
Nov
20
comment If $\alpha$ is of bounded variation on $[a,b]$, then it is continuous almost everywhere on $[a,b]$
Very helpful! Thank you.
Nov
20
comment If $\alpha$ is of bounded variation on $[a,b]$, then it is continuous almost everywhere on $[a,b]$
I guess I thought my attempt was an "outline" and that there might be a better way to state the proof with variables/numbers.
Nov
19
comment Explicit formula of Mobius Transformation that maps non intersecting circles to concentric circles
what if the circles are nested to begin with? Like $|z|=1$ and $|z-1/4| = 1/4$. These are nested but not concentric.
Nov
18
comment Show $ \int_{C_N} \frac{\pi \sec(\pi z)}{z^3} dz \to 0$ for given contour.
Yes, I have no trouble computing the integral. How do I use $|\cos \pi z|^2$ to find a lower bound? I've been working on this for a while, and have gotten stuck on this (seemingly) simple part of the proof.