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Dec
11
awarded  Promoter
Dec
9
accepted Without using beta function, for what $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge?
Dec
9
awarded  Caucus
Dec
9
asked Describe the Riemann surface for $w=z^2-1$.
Dec
9
asked Without using beta function, for what $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge?
Dec
9
accepted For what values $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge.
Dec
9
accepted Show the level curves of $\log|f(z)|$ are orthogonal to those of $\operatorname{arg}(f(z))$.
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
8
asked Show the level curves of $\log|f(z)|$ are orthogonal to those of $\operatorname{arg}(f(z))$.
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
7
asked For what values $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge.
Dec
6
accepted 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?
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
6
asked 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?
Dec
4
accepted Prove $f'(x) \geq x f(x)$ $\forall x \in \mathbb{R}$ $\implies$ $\exists k$ s.t. $ke^{x} \leq f(x)$ $\forall x \in \mathbb{R}$.
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!
Dec
2
accepted $\{f_n\} \in \mathcal{R}[a,b]$, converges pointwise to $f\in \mathcal{R}[a,b]$. Prove $\lim_{n\to\infty}\int_a^b f_n \, dx = \int_a^b f \, dx$.
Dec
2
revised $\{f_n\} \in \mathcal{R}[a,b]$, converges pointwise to $f\in \mathcal{R}[a,b]$. Prove $\lim_{n\to\infty}\int_a^b f_n \, dx = \int_a^b f \, dx$.
[Edit removed during grace period]
Dec
2
accepted If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$