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Apr
18
answered Book recommendation for Linear algebra.
Apr
18
comment Book recommendation for Linear algebra.
Strang, whose book s mentioned below, has free video lectures of his course at MIT Open Courseware. This is the more "practical" LA course at MIT. On the more theoretical side, They tend to use Axler, "Linear Algebra Done Right."
Apr
18
comment Book recommendation for Linear algebra.
Have you searched for similar questions here? They have been answered with numerous, well-detailed recommendations.
Apr
2
comment Nice book on geometry to gift an undergraduate in mathematics
Thanks - very cool!
Apr
2
comment Nice book on geometry to gift an undergraduate in mathematics
Avatamsaka Sutra
Mar
19
comment Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$
For another perspective on the relation between the PNT and Re $\rho$ < $1$, you might want to look at page 68 of Edwards, "Riemann Zeta Function."
Mar
11
comment Zeta function for negative integers
Have a look at Stopple "Primer in Analytic Number Theory" Section 8.2, especially the problems and solutions in the back. Or Edwards "Riemann Zeta Function" Section 1.5
Mar
9
revised Best Sets of Lecture Notes and Articles
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Feb
19
comment Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
Thanks Ron. So in the first integral in the denominator, I have $1 - 1 + 2\pi \epsilon e^{i\phi}$ + higher order terms. Then the $e^{i \phi}$ in the numerator would cancel the one in the first order term. For the rest of the numerator, I would have $1 + ia\epsilon e^{i\Phi}$ + higher order terms. So I am stuck how to deal with the higher order terms, and what happens to the $a$. Thanks, sorry to be so dense.
Feb
19
comment Integrate: $\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)}dx$
Dear Ron, I saw this on your blog as well. Didn't know where you would prefer to be asked a question (maybe neither - but hope not since you are always so generous). Sorry to be a serial questioner, yet your work is so compelling. Could I please ask you how you applied the residue theorem to get from the fifth and sixth integrals to the "combine to form" line. On second thought, this is the better place to ask so I can +50 for this and your untiring generosity in helping others. All the best, Andrew
Feb
19
comment Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
Thanks for you help. All the best, Andrew
Feb
18
comment Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
so am unable to get his results as well. Thanks very much for your consideration. All the best,
Feb
18
comment Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
Dear Raymond, Maybe I could please ask you to elaborate as to how you get (2) & (3). I know you showed how for the half circle you get "half" a residue here math.stackexchange.com/questions/478534/…. But that example is in a nice convenient form with the $\frac{C_{-1}}{z-z_0}$ of a Laurent series. So how do you get the terms in the ( ) in each of (2) and (3)? I tried using $z = \epsilon e^{i \theta}$, but that got pretty messy and is far from the simplicity of your terms. Also looked at the link in Ron's blog where he uses $z$ as I mentioned
Feb
18
comment Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
Thanks, Ron. As you know, you are a key All-Pro on my fantasy math team. All the best, Andrew
Feb
18
comment Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$
Dear Ron, Could I please trouble you for a hint as to what looks how to isolate $e^{-kx}$ outside the integral so you have $\Gamma(u)$ times the summation. Actually I fear that posing this question is naive. But that's what it looks like. Actually, probably much better is a hint how to get from the second expression to the third. Thanks very much. With regards, Andrew
Feb
17
comment Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$
Dear Raymond - Outstanding. Beautifully resolves the "Update" conjecture. I was wondering if it also addresses the initial conjecture(4)? I.e., does the update come out of (4), or are they distinct? Thanks and best regards,
Feb
16
comment Evaluating $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$
Nice generalization above - thanks very much. Best,
Feb
15
comment Evaluating $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$
Dear Raymond, Maybe you would please elaborate a bit how to get from the line above "As for the multiplication theorem" to the last line. Thanks and all the best,
Feb
12
comment How to prove Gauss’s Multiplication Formula?
@JacobMayle Me too, a big help.
Feb
10
revised Information about Riemann Zeta function
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