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Wrong numerical value

Posted 9 years ago
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Dear people of the wolfram community,

I'm reproducing results from calculations done in published physics papers.

The paper states:

Sqrt[2] EllipticE[1/Sqrt[2]] - EllipticK[1/Sqrt[2]]/Sqrt[2] = Sqrt[2] Pi^(3/2) / Gamma[1/4]^2 is approximately: 0.56.

However if I try:

N[Sqrt[2] EllipticE[1/Sqrt[2]] - EllipticK[1/Sqrt[2]]/Sqrt[2]]

I find the value: 0.274974.

Am I doing something wrong? Is the value of 0.56 correct?

I would be very grateful to anyone who can answer this question.

M.

P.S.: The paper in question is: "Wilson-Polyakov Loop at Finite Temperature in Large N Gauge Theory and Anti-de Sitter Supergravity". Published in Nucl.Phys; Paper on Arxiv; has over 200 citations.

POSTED BY: M. R.
3 Replies

Sorry for the double posting. I was still typing and did not see that there was already a reply.

M.

POSTED BY: Marco Thiel

Dear M.R.

There might be a little problem with the definitions there. Note the following two websites:

http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html

http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html

You will see that they say:

It is implemented in Mathematica as EllipticK[m], where m=k^2 is the parameter.

It is implemented in Mathematica as EllipticE[m], where > m=k^2 is the parameter.

Now, if we calculate

N[Sqrt[2] EllipticE[1/Sqrt[2]^2] - EllipticK[1/Sqrt[2]^2]/Sqrt[2]]

that gives 0.59907, which is consistent in part with the manuscript Eq. (36) which says that it equals the expression

Sqrt[2] Pi^(3/2)/(Gamma[1/4])^2 // N

which is also 0.59907. This is already closer to the value of 0.56 that they use in the manuscript.

I see that in the paper that was published in 1998 in Nuclear Physics B: S. Rey et all, "Wilson-Polyakov Loop at Finite Temperature in Large N Gauge Theory and Anti-de Sitter Supergravity", Nuclear Physics B 527 (1998), 171- 186 they also give a numerical value of 0.56.

I am not sure whether this goes in the right direction or not; and/or whether it helps.

Cheers, Marco

POSTED BY: Marco Thiel

First I checked the value of

Sqrt[2] Pi^(3/2) / Gamma[1/4]^2

which actually evaluated to:

0.59907

which is not quite what the paper says. I also noticed that the paper is using K and E to denote the complete elliptic integrals of the first and second kind respectively. Unfortunately elliptic integrals can have different notations depending on whose definition is being used. In particular Mathematica (and Abramowitz and Stegun) uses:

$$K (m)=\int _0^{\pi/2}\frac{1}{\sqrt{1-m \sin^2\theta}} d \theta$$

while some others use:

$$K (m)=\int _0^{\pi/2}\frac{1}{\sqrt{1-m^2 \sin^2\theta}} d \theta$$

I assumed that the difference you get is likely due to this, so I tested it (note the squared argument now):

N[Sqrt[2] EllipticE[(1/Sqrt[2])^2] - EllipticK[(1/Sqrt[2])^2]/Sqrt[2]]

which also returns:

0.59907

For clarity of course, I would recommend contacting the authors to determine which definition of the elliptic integrals they used (it would have been nice if they included to reference for it).

POSTED BY: Jesus Hernandez
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