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comment $L$-function, easiest way to see the following sum?
For an algebraic method see here, for an analytic solution see here.
Jun
28
comment How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?
The paper by Gourdon and Sebah "Numerical evaluation of the Riemann Zeta-function" and this thread may help.
Jun
22
comment Numerical precision of arctan function
@YvesDaoust: I was just adding the same suggestion (for the case $x^2+y^2\ll z^2$ at least)... :-)
Jun
22
comment Numerical precision of arctan function
(answer converted in a comment) Concerning $\theta$ and $\phi$ the expressions should be : $$\theta = \arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)\\ \phi = \arctan\left(\frac yx\right)$$ (the last one by dividing $y$ and $x$, the first one using $\cos(\theta)=\dfrac zr$). Supposing your code correct the problem was probably explained by Victor Liu : $\;\arccos\,$ may have a large relative error while evaluating $\arccos(1-\epsilon)$. Computing acos(1-0.5e-8)*1e4, acos(1-0.5e-10)*1e5, acos(1-0.5e-12)*1e6 using gcc on a mac I obtained 0.9999999974, 1.0000000414, 1.0000444493 0.9996002812.
Jun
13
comment How can i represent 3D space using 4x4 matrix?
See for example here for the direct generalization.
Jun
10
comment How to check if a point is inside a rectangle?
Shortly yes @bgoers. We impose the perpendicular projection of $\textbf{M}$ on $\textbf{AB}\;$ to be between $\textbf{A}$ and $\textbf{B}$ and the perpendicular projection of $\textbf{M}$ on $\textbf{AD}\;$ to be between $\textbf{A}$ and $\textbf{D}$. The result will follow from the condition that $\textbf{AB}\;$ and $\textbf{AD}\;$ are perpendicular. This condition is thus required for this method to work as well as accurate positions (floating points values instead of nearest integers) for the positions. Fine continuation,
Jun
10
comment Continuous function from $(0,1)$ onto $[0,1]$
The link was broken.
May
29
awarded  Nice Answer
May
28
comment Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$
You may be interested by Berndt's "Ramanujan's notebook III" !page $163$ and !page $164$. Concerning Roger's $1907$ paper you may read/download it at archive.org. Start with Berndt's page $164$ substitutions to rewrite the c.f. in the p.$163$ form (I didn't reverify all this...). Hoping this helped a little,
May
24
comment Integral formulas involving continued fractions
for (I think) any positive real $n$. Alternative expressions with $\psi$ the digamma function : $$\int_0^\infty \frac{e^{-n\,t}}{\cosh(t)} dt=\frac 12\left[\psi\left(\frac {3+n}4\right)-\psi\left(\frac {1+n}4\right)\right]$$
May
24
comment Integral formulas involving continued fractions
$(2)$ should be given by (from a slightly more general formula by Rogers according to my notes) : $$\dfrac{1}{n +}\dfrac{1^{2}}{n +}\dfrac{2^{2}}{n +}\dfrac{3^{2}}{n + \cdots}=2\sum_{k=1}^\infty \frac {(-1)^{k+1}}{n+2k-1}$$
May
23
comment How to avoid stupid mistakes in calculus exams without checking the whole process?
Barry Cipra's fun little book "Misteaks. . . and how to find them before the teacher does" may be helpful here
May
22
comment Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$
Related to Cortizol's comment and Dr. MV's answer see this thread.
May
22
comment Is there any relationship between the Riemann z function and strange attractors?
Two more things you may enjoy : the universality of zeta and the related paper by Woon "Riemann zeta function is a fractal". Fine reading,
May
21
awarded  Nice Answer
May
20
comment Value of polylogarithms $\mathrm{Li}_s(1)$ for $s<1$.
Yes @gammatester. From the equality for $\Re s >1$ and since $\zeta$ admits a unique analytic continuation over the whole complex plane except $s=1$ we must have the equality for $s\neq 1$ (else $\operatorname{Li}_s(1)$ would not be analytic in $s$ !).
May
20
comment About Mertens' first theorem
Nice to meet you here @draks! An answer would require more work (at least for me) to find something simple, compare the solutions and why not provide a proof... :-) (but I am too busy !). Cheers,
May
20
comment About Mertens' first theorem
Glad it helped @wiskundeliefhebber! I don't know a simple proof of this (which may involve a goot part of the material needed for PNT). Fine continuation,
May
20
comment About Mertens' first theorem
See $(17)$ here and the links concerning $B_3$ for example at OEIS.
May
20
revised Value of polylogarithms $\mathrm{Li}_s(1)$ for $s<1$.
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