
MHB Craftsman
#1
June 4th, 2014,
20:48
Hi all!
I'll try and be brief yet concise (the former won't happen, and the jury's out on the latter), but please bear with me... Not to mention wish me luck, there's a dear. Thanks!
I'm currently learning Cprogramming, and as such, am in the process of designing a number of 'portable', mathematical, approximation tools (simple Cprograms anyone could download and run on their compooter). In short, simple programs for, say, approximating the value of certain higher functions, like the Gamma function, Clausen function, Polygamma function, Barnes' Gfunction, and the Howmuchisthatdoggyinthewindow function (Hmm!).
On that note, here's a few random Q's:
 Since a lot of folk on here likely don't have fancy tools like Mathematica  myself included  would/could there (a) be a place to upload a few such simple tools on here, for others to use, and (b) would there be a demand for such things to begin with?
Well actually, that's it. That's all my Q's spent, and all in one go, no less. I feel quite poor now. Intellectually skint.
One other thing... (yay! I found my mental wallet again!):
Depending on the type and complexity of the functions involved  and I'd have to have a basic grasp of the function and branch of mathematics in question  I'd be up for doing a few bespoke programs, as per forum members (potential) requests.
To give just one (obscure) example, I'm currently quite interested in limits of infinite square roots, possibly alternating, so, to sate my own particular mathematical fetishes, I'm currently working on programs to approximate the following functions:
$ \displaystyle \mathscr{P}(x) = \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } }$
$ \displaystyle \mathscr{Q}(x,y) = \sqrt{x+ \sqrt{y+ \sqrt{x+ \sqrt{y+ \sqrt{x+ \cdots } } } } }$
$ \displaystyle \mathscr{R}(x,y) = \sqrt{x+ \sqrt{y \sqrt{x+ \sqrt{y \sqrt{x+ \cdots } } } } }$
I'll also be doing quite a few programs for varyingdegree precision calculations of infinite series and infinite products. Such programs, once a template is made, could easily be adapted for things like, say, the Riemann Zeta function $ \displaystyle \zeta(x)$, to name but one example. If the place to upload them is there, and the demand is there too, I'd be more than happy to take requests, and build small, portable, bespoke programs (as and when time permits).
Up to you guys and gals...
All the best!!
Gethin
Last edited by DreamWeaver; June 4th, 2014 at 20:52.
Reason:
improve thread title readability (sorry)

June 4th, 2014 20:48
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#2
June 4th, 2014,
22:14
I say go for it.
Incidentally, with the first one, you can do this:
\begin{align*}
y&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } } \\
y^2&=x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2  x&= \sqrt{x+ \sqrt{x+ \sqrt{x+ \sqrt{x+ \cdots } } } } \\
y^2x&=y \\
y^2  y  x&=0 \\
y&=\frac{1\pm \sqrt{1+4x}}{2},
\end{align*}
for $1+4x\ge 0$.

MHB Craftsman
#3
June 5th, 2014,
10:30
Thread Author

#4
June 5th, 2014,
15:49
You can do a similar thing for the other two, but you get a quartic. Mathematica spits out the answer pretty fast, but it's quite complicated.

MHB Craftsman
#5
June 5th, 2014,
16:02
Thread Author
Originally Posted by
Ackbach
You can do a similar thing for the other two, but you get a quartic. Mathematica spits out the answer pretty fast, but it's quite complicated.
Aha! Thanks for that, Ackbach!! Alas, I don't have Mathematica, so that's why I was thinking of creating such tools, for those poor souls like myself without them.
More generally  just on this one particular type of functions  the examples given above were of the more basic kind. The general case I'm examining is
$ \displaystyle \Omega_{\infty}(s,x) = \Bigg[ \Bigg[ \Bigg[ \Bigg[ \cdots \, + x \Bigg]^s \, + x \Bigg]^s \, + x \Bigg]^s \, + x \Bigg]^s $