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Dec
13
comment I like math, but can't keep up with the pace.
You will fail to get the required intuition at times, yes. But you have to try again after.
Dec
13
comment I like math, but can't keep up with the pace.
...and why are you so afraid of sinking? Can you not try again after?
Dec
13
comment Is there a means of finding an infinite sum by means of altering it into an integral?
The other possibility is to express the sum as a contour integral, with the summand multiplied by a factor with poles at the right places.
Dec
13
comment Is there a means of finding an infinite sum by means of altering it into an integral?
Alternatively, look up the Abel-Plana formula.
Dec
12
comment Sum of the Stieltjes constants? (divergent summation)
Then, it is possible that PARI/GP's algorithms are unable to handle this. In Mathematica, I used the double exponential quadrature method of Takahasi and Mori.
Dec
12
comment Generalization for Stirling numbers 2nd kind to negative column-indexes?
So, doesn't that reflection relation answer the question in your title?
Dec
12
comment What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?
@Kirill, are you sure of all those digits you quoted? My own experiments of summation + extrapolation only seem to guarantee the first six significant digits of your result, which makes me suspicious of your other digits.
Dec
12
comment Sum of the Stieltjes constants? (divergent summation)
Have you tried taking the imaginary part before integrating?
Dec
12
comment Generalization for Stirling numbers 2nd kind to negative column-indexes?
Have you seen this? Formula 2.4 there might be of interest to you.
Dec
12
comment What is the half-derivative of zeta at $s=0$ (and how to compute it)?
I'll link to a previous thread of yours, where I derived expressions that will pop up in the Maclaurin series for $\zeta(s)$. Semidifferentiate this series, replace $s$ with $0$, and you now have a series to start with. I might write something up late if I find the time.
Dec
12
comment What is locus of a fixed point on a circle of radius $r$ rolling over the curve $y=\sin x$?
I used Mathematica to generate the cartoon, @marty.
Dec
11
comment Characterizations of cycloid
@Blue, have a look at the cartoon in my answer. :)
Dec
11
comment What is the length of a sine wave from $0$ to $2\pi$?
@Sol, we will have to agree to disagree, then; $\pi$ just happens to be a convenient constant for me, as opposed to a function that may or may not be implemented in a given computing environment.
Dec
11
comment What is the length of a sine wave from $0$ to $2\pi$?
@Sol, really, you consider $r \theta$ to be an "infinite series"?
Dec
11
comment Circle Rolling on Ellipse
As I noted in the comments, the elliptic integral of the second kind is necessary here, since it pops up in the expression for the arclength of the ellipse. So yes, complicated…
Dec
11
comment Closed form for $\sum_{n=1}^\infty \frac 1 {2^n - 1}$
Have a look at the Lambert series.
Dec
11
comment Roll one ellipse on another: Locus of center ever a circle?
Good thing I found this question, then. :)
Dec
10
comment Roll one ellipse on another: Locus of center ever a circle?
Bonus: the usual lemniscate of Bernoulli can be obtained as the trace of the center of an equilateral hyperbola rolling on a congruent hyperbola. (In fact, the Booth lemniscates in general are defined as the pedal curves of a central conic.)
Dec
10
comment Roll one ellipse on another: Locus of center ever a circle?
Its parametric equations will certainly involve the incomplete elliptic integral of the second kind; that much is certain.
Dec
10
comment Roll one ellipse on another: Locus of center ever a circle?
@Tony, actually, nothing in the usual definition of a "roulette" says that the point should be on the circumference. Thus, both cycloids and trochoids are considered roulettes, to use the classical example.