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23m
comment Proving Plancherel's theorem using Cauchy integral formula
Great work $+1$
23m
comment Proving Plancherel's theorem using Cauchy integral formula
@Batominovski has filled in the details you missed but let me caution you about one thing: going from line two to line three you made two very critical assumptions: 1) you can replace an improper integral with a Cauchy principal value integral and 2) that you can pull the $k_0$ limit outside the $x'$ integral. These are very hefty assumptions so you have not truly proved Plancherel's theorem in full generality, only in a very specific situation. That situation being when both of those very big assumptions hold.
30m
comment Proving that a sequence converges or diverges
I edited your post. Please verify that this is what you meant.
30m
revised Proving that a sequence converges or diverges
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32m
comment Proving Plancherel's theorem using Cauchy integral formula
Wouldn't that change the argument of $f$?
38m
comment Proving Plancherel's theorem using Cauchy integral formula
I'm curious about how you simplified the two exponentials to one.
1h
comment Definition for the set of Real Numbers
Hah I was typing something similar to this. $+1$
2h
comment Let T be the bounded operator and T* be the adjoint operator of T.Show the following.
Welcome to Math Stack Exchange! Your post is being voted to close for two reasons: 1) your post is incredibly difficult to read; 2) you have not shown any effort on your end. We do not mind helping out here (or even giving full answers sometimes) if sufficient work is shown on behalf of the poster. In order to prevent your post from being closed, you should show some work and clean up your question considerably. If you are unable to answer these questions because you're fundamentally struggling with the notion of adjoints, then perhaps you should ask a new question for clarification on that.
2h
comment Necessary to assume $f\in C^\infty$ in this Fourier transform problem?
It definitely does seem to be rather loosely put together. Either that or the authors had something very different (read: very laborious) in mind.
2h
comment Necessary to assume $f\in C^\infty$ in this Fourier transform problem?
What I find interesting is that (seemingly) you could get away with a weaker statement: replace $\xi^{1+\varepsilon}$ with $\xi^{\frac{1}{2}+\varepsilon}$. Assuming your analysis is correct - which it seems to be.
6h
revised When and why can functions “take on” the role of vectors in defining vector spaces?
edited title
11h
comment How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.
See here: en.wikipedia.org/wiki/Integral_of_inverse_functions There is a very beautiful way to view this graphically.
12h
revised Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer?
edited tags
12h
revised Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer?
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12h
revised Is $\left(45+29\sqrt{2}\right)^{1/3} + \left(45-29\sqrt{2}\right)^{1/3}$ an integer?
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1d
revised Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$
edited title
1d
comment Undergraduate set theory research
The issue then still remains that how are you going to actually show you've done any sort of advanced work in set theory? You'll have nothing to substantiate your claims.
1d
comment Undergraduate set theory research
You should really consult someone in your department about this. It is hard for us to know exactly what level you are at and how much you can handle. Set theory becomes incredibly challenging beyond foundations. Moreover, just because you do an independent research project doesn't mean anything. You will have nothing to show for it on applications if you do not work under a professor (except your say-so on the matter).
1d
awarded  Nice Answer
1d
comment Must all Lebesgue integrable functions really be invertible?
By the way, you should accept one of the answers to this question if you feel they have appropriately answered your question! I see that you don't accept a lot of answers to your questions. This is a bad habit and can leave people with the feeling that they're being used. Moreover, questions with no accepted answers will be bumped by Community from time to time.