[math-fun] (Log[2]-5/8) = 0.06814718 . . .
For a horrible approximation of Pi/2, and a minor headache along the way: Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653 Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left? After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less). Am I missing something? Is this (Log[2]-5/8) somehow easy to derive? --Brad See also: http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/
I'm not a mathematician, so if this question is out of line, just ignore it. But I wonder at the use of a recursive computer program or infinite series to approximate some transcendental number, that uses, in the program, an irrational or even a transcendental function. Isn't the point of approximation programs to use only rational operations to achieve the approximation? Brent On 1/24/2020 8:19 PM, Brad Klee wrote:
For a horrible approximation of Pi/2, and a minor headache along the way:
Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653
Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left?
After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less).
Am I missing something? Is this (Log[2]-5/8) somehow easy to derive?
--Brad
See also:
http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/
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As I think you can tell by my recent failure to figure out how Mathematica works, I am no mathematician either. Anyways, there are a lot of reasons that we would want to produce approximations, and only sometimes would I care if they were rational numbers or not. In this case, I simply want to reverse EllipticE[x], so that I can evaluate more rapidly when x>0.5, as shown in this graph: RevElE = Normal[ERev2 + c*ERev1 /. Solve[ (Normal[N[ERev2 + c*ERev1 - Pi/2] ] /. x -> 1) == 0, c][[1]]]; RevElE2 = Normal[ERev2 + (Log[2] - 5/8)*ERev1]; LogPlot[{ Abs[RevElE - EllipticE[1 - x]], Abs[RevElE2 - EllipticE[1 - x]] }, {x, 0, 1}, PlotStyle -> {Red, Blue}] Out[] = https://0x0.st/irJo.png The blue curve reaches machine precision, while the red curve has serious error due to poor solving of the constant "c", which should equal (Log[2] - 5/8) ???? Even then, the approximations toward EllipticE[0] turn out to be nearly rational, compare the following: Expand[Normal[Series[ERev2 + (Log[2] - 5/8)*ERev1 - Pi/2, {x, 0, #}]] /. x -> 1 & /@ Range[5]] // TableForm 3/4 - Pi/2 + Log[2] 35/64 - Pi/2 + 11*Log[2])/8 etc... all of the form R1 + R2*Pi + R3*Log[2]. This is not too bad, but also "horrible" and a "headache". Especially after hours of computer hacking and still no explanation for the occurrence of a rogue Log[2]???? Beware of the Radiance, --Brad On Fri, Jan 24, 2020 at 10:32 PM Brent Meeker via math-fun <math-fun@mailman.xmission.com> wrote:
I'm not a mathematician, so if this question is out of line, just ignore it. But I wonder at the use of a recursive computer program or infinite series to approximate some transcendental number, that uses, in the program, an irrational or even a transcendental function. Isn't the point of approximation programs to use only rational operations to achieve the approximation?
Brent
On 1/24/2020 8:19 PM, Brad Klee wrote:
For a horrible approximation of Pi/2, and a minor headache along the way:
Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653
Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left?
After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less).
Am I missing something? Is this (Log[2]-5/8) somehow easy to derive?
--Brad
See also:
http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/
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participants (2)
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Brad Klee -
Brent Meeker