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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?
Nov
27
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.
Nov
27
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 $(1+x^2)$?
Nov
27
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.
Nov
27
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?
Nov
27
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.
Nov
26
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 $\sin$?