Cool! I tried ries on two of my favorite numbers (arising from theory of voting via some difficult integrals which might have a closed form but I used monte carlo) 0.2586333(3) = 0.2586335(4) and 0.1300873(4) = 0.1300876(2) [standard errors in last digit shown in parens] but it didn't find anything I liked. The correct formula (if there is one) may involve erf, e, pi, ln, and algebraic numbers. 0.1300876 = 1/(ln(2)*phi^5) [??] is suitably mysterious, I admit, and does match my accuracy. Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Based on the manual, it looks like special functions (erf) aren't included. You might be able to add them.
Actually, I practically never understand how Gosper gets his formulas. Many of us feel the same way.
Rich --- Quoting Warren Smith <warren.wds@gmail.com>:
Cool! I tried ries on two of my favorite numbers (arising from theory of voting via some difficult integrals which might have a closed form but I used monte carlo) 0.2586333(3) = 0.2586335(4) and 0.1300873(4) = 0.1300876(2) [standard errors in last digit shown in parens] but it didn't find anything I liked. The correct formula (if there is one) may involve erf, e, pi, ln, and algebraic numbers.
0.1300876 = 1/(ln(2)*phi^5) [??] is suitably mysterious, I admit, and does match my accuracy.
Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
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Hi all, Just came across/worked out something and the result was interesting at least to me anyway even though it's something pretty simple ;) So here's a quick challenge question: What's special about the cube root of 4 relating to areas ? bye Dave
Ummm... Amount of wrapping paper to wrap a double-size box. If you double the size of a cube, its edge length goes up by the cube root of 2, and therefore its surface area (and thus, the amount of paper required to wrap it) increases by the cube root of 4. The same thing applies to other shapes so long as the doubled shape is similar (in the Euclidean-geometrical sense of the word) to the original, and isn't one of those monsters that all the kids (like Helge and Georg) keep going on about. Or, if you want something more approximate, how about 3/(e-1) + sin(1) - 1 = 1.5874011054... Is that it? (-: - Robert On Tue, Dec 27, 2011 at 19:03, David Makin <makinmagic@tiscali.co.uk> wrote:
Hi all,
Just came across/worked out something and the result was interesting at least to me anyway even though it's something pretty simple ;) So here's a quick challenge question:
What's special about the cube root of 4 relating to areas ?
bye Dave
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Hi, *****
If you double the size of a cube, its edge length goes up by the cube root of 2, and therefore its surface area (and thus, the amount of paper required to wrap it) increases by the cube root of 4. The same thing applies to other shapes so long as the doubled shape is similar (in the Euclidean-geometrical sense of the word) to the original, and isn't one of those monsters that all the kids (like Helge and Georg) keep going on about.
*** Now that's also pretty interesting :) The answer I was looking for was the value of x such that the area under y=x^2 is equal to the area under y=sqrt(x) between the origin and x (area=4/3). Because I'm very paranoid and a little rusty when it comes to actually performing such calculations I checked it online via this rather nice tool: http://newgraph.seriesmathstudy.com/ On 28 Dec 2011, at 05:14, Robert Munafo wrote:
Ummm... Amount of wrapping paper to wrap a double-size box.
If you double the size of a cube, its edge length goes up by the cube root of 2, and therefore its surface area (and thus, the amount of paper required to wrap it) increases by the cube root of 4. The same thing applies to other shapes so long as the doubled shape is similar (in the Euclidean-geometrical sense of the word) to the original, and isn't one of those monsters that all the kids (like Helge and Georg) keep going on about.
Or, if you want something more approximate, how about 3/(e-1) + sin(1) - 1 = 1.5874011054...
Is that it? (-:
- Robert
On Tue, Dec 27, 2011 at 19:03, David Makin <makinmagic@tiscali.co.uk> wrote:
Hi all,
Just came across/worked out something and the result was interesting at least to me anyway even though it's something pretty simple ;) So here's a quick challenge question:
What's special about the cube root of 4 relating to areas ?
bye Dave
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Oh yeah, that too. I think I ran across that once, it looks like a few calculus problems I may have seen... I guess I've been wrapping too many gifts recently. Kinda got stuck on the physical notion of cube-roots and areas (-: And of course, I should have said "double the volume of..." rather than "double the size of...". Size matters, but volume is more specific. - Robert On Wed, Dec 28, 2011 at 04:59, David Makin <makinmagic@tiscali.co.uk> wrote:
Hi,
*****
If you double the size of a cube, its edge length goes up by the cube root of 2, and therefore its surface area (and thus, the amount of paper required to wrap it) increases by the cube root of 4. The same thing applies to other shapes so long as the doubled shape is similar (in the Euclidean-geometrical sense of the word) to the original, and isn't one of those monsters that all the kids (like Helge and Georg) keep going on about.
***
Now that's also pretty interesting :)
The answer I was looking for was the value of x such that the area under y=x^2 is equal to the area under y=sqrt(x) between the origin and x (area=4/3).
Because I'm very paranoid and a little rusty when it comes to actually performing such calculations I checked it online via this rather nice tool:
Using maxima I get three answers, two of them complex: (%i4) solve(integrate(x^2, x, 0, y) = integrate(sqrt(x), x, 0, y), y); Is y positive, negative, or zero? positive; (sqrt(3) %i - 1) sqrt(y) (sqrt(3) %i + 1) sqrt(y) (%o4) [y = ------------------------, y = - ------------------------, 2/3 2/3 2 2 1/3 y = 2 sqrt(y)] All of them are in the form y=K sqrt(y), which is easy to solve manually but annoying. I cannot afford Mathematica. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
* Warren Smith <warren.wds@gmail.com> [Dec 28. 2011 18:30]:
[...]
Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Just as quick copy and paste (suggest to start with the last one): J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.} Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.} Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.} Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.}
Hello, the formula for (1/4)! is quite interesting, the approximation is 88 digits, this is unique. Now about Gamma(n/48), I do not think that any higher value would lead to simple approximations as mr Gosper showed, as the index increases : it gets quite messy, we can only hope to get the first denominators in my opinion. Nevertheless, the values for (1/3)! and (1/4)! are impressive, the technique uses <approximations> of dedekind functions, a neat trick. best regards, Simon Plouffe Le 28/12/2011 18:43, Joerg Arndt a écrit :
* Warren Smith<warren.wds@gmail.com> [Dec 28. 2011 18:30]:
[...]
Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Just as quick copy and paste (suggest to start with the last one):
J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.}
Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.}
Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.}
Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.}
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On the subject of the gamma function, I'd be interested to know if I got this correct for complex gamma - it's the func I wrote for the Ultra Fractal math library = obviously it's not the most accurate implementation but I was balancing speed with accuracy and in adition it was taking too long to hunt down a better table of values to use ;) *************** ; Complex Gamma() David Makin June 2008 ; @param pz complex argument ; @return Gamma function of argument static complex func ComplexGamma(complex pz) complex w = pz if pz==0.0 w = recip(0) else if real(pz)<0.0 w = -pz if imag(pz)==0.0 if (real(pz)%1)==0.0 w = recip(0) endif endif endif if real(w)<1e300 w = (sqrt(2.0*#pi)* \ (0.99999999999999709182 \ + 57.156235665862923517/(w+1.0) \ - 59.597960355475491248/(w+2.0) \ + 14.136097974741747174/(w+3.0) \ - 0.49191381609762019978/(w+4.0) \ + .33994649984811888699E-4/(w+5.0) \ + .46523628927048575665E-4/(w+6.0) \ - .98374475304879564677e-4/(w+7.0) \ + .15808870322491248884E-3/(w+8.0) \ - .21026444172410488319e-3/(w+9.0) \ + .21743961811521264320e-3/(w+10.0) \ - .16431810653676389022e-3/(w+11.0) \ + .84418223983852743293e-4/(w+12.0) \ - .26190838401581408670e-4/(w+13.0) \ + .36899182659531622704e-5/(w+14.0)) \ /w) * exp(-w-5.2421875)*(w+5.2421875)^(w+0.5) if real(pz)<0.0 w = pz*w*sin(#pi*pz) if (w==0.0) w = recip(0) else w = -#pi/w endif endif endif endif return w endfunc ************** On 28 Dec 2011, at 18:13, Simon Plouffe wrote:
Hello,
the formula for (1/4)! is quite interesting, the approximation is 88 digits, this is unique.
Now about Gamma(n/48), I do not think that any higher value would lead to simple approximations as mr Gosper showed, as the index increases : it gets quite messy, we can only hope to get the first denominators in my opinion.
Nevertheless, the values for (1/3)! and (1/4)! are impressive, the technique uses <approximations> of dedekind functions, a neat trick.
best regards, Simon Plouffe
Le 28/12/2011 18:43, Joerg Arndt a écrit :
* Warren Smith<warren.wds@gmail.com> [Dec 28. 2011 18:30]:
[...]
Then I tried Gosper's (1/4)! and I don't understand how Gosper got his formula. Actually, I practically never understand how Gosper gets his formulas. As I mentioned before, it'd be nice to find formulas for GAMMA(k/48) for example. Such formulas might exist in terms of algebraic numbers and the AGM.
Just as quick copy and paste (suggest to start with the last one):
J.\ M.\ Borwein, I.\ J.\ Zucker: {Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind}, IMA Journal of Numerical Analysis, vol.12, no.4, pp.519-526, \bdate{1992}.}
Greg Martin: {A product of Gamma function values at fractions with the same denominator}, arXiv:0907.4384v1 [math.CA], \bdate{24-July-2009}. URL: \url{http://arxiv.org/abs/0907.4384}.}
Albert Nijenhuis: {Small Gamma Products with Simple Values}, arXiv:0907.1689v1 [math.CA], \bdate{9-July-2009}. URL: \url{http://arxiv.org/abs/0907.1689}.}
Raimundas Vid\={u}nas: {Expressions for values of the gamma function}, arXiv:math.CA/0403510, \bdate{30-March-2004}. URL: \url{http://arxiv.org/abs/math/0403510}.}
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participants (7)
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Dave Makin -
David Makin -
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Robert Munafo -
Simon Plouffe -
Warren Smith