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%I #40 May 08 2018 15:11:56
%S 4,5,3,2,3,6,0,1,4,1,8,2,7,1,9,3,8,0,9,6,2,7,6,8,2,9,4,5,7,1,6,6,6,6,
%T 8,1,0,1,7,1,8,6,1,4,6,7,7,2,3,7,9,5,5,8,4,1,8,6,0,1,6,5,4,7,9,4,0,6,
%U 0,0,9,5,3,7,2,1,3,0,5,1,0,2,2,5,9,0,8,3,8,7,9,6,0,4,0,1,6,0,8,9,6,5,3
%N Decimal expansion of Van der Pauw's constant = Pi/log(2).
%C Van der Pauw developed a method for measuring the sheet resistance of a four-terminal conducting sheet of arbitrary shape. Assuming the terminals to be point contacts at the periphery of the structure, he proved a general theorem that yields an analytical expression for the sheet resistance Rs. In the special case that the structure is invariant for a rotation of ninety degrees, the formula of Van der Pauw is Rs = (Pi/log(2))*(V/I).
%C A general theorem for the sheet resistance Rs of a Van der Pauw structure with finite contacts that is invariant for a rotation of ninety degrees was proved by Versnel. His theorem states that Rs = [K(k1)/K'(k1) - K(k2)/(2*K'(k2))]^(-1)*(V/I) with K(k) and K'(k) complete elliptic integrals with modulus k (Abramowitz and Stegun use parameter m = k^2).
%C Versnel found, with a little help from the author, expressions for Rs = C(d)*(V/I) for several Van der Pauw structures if d, the ratio of the sum of the lengths of the contacts and the length of the boundary of the sheet, tends to zero, see the formulas (first two terms are given). For point contacts, i.e., d = 0, Van der Pauw's constant appears.
%H G. C. Greubel, <a href="/A163973/b163973.txt">Table of n, a(n) for n = 1..5000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 17, pp. 589-626.
%H L.J. van der Pauw, <a href="https://es.scribd.com/doc/44653007/1958-Van-Der-Pauw-Philips-Res-Rep-a-Method-of-Measuring-Specific-Resistivity-and-Hall-Effect-of-Discs-of-Arbitrary-Shape#scribd">A method of measuring specific resistivity and Hall effect of disc of arbitrary shape</a>, Philips Research Reports, Vol. 13. no. 1, pp 1-9, February 1958.
%H W. Versnel, <a href="http://dx.doi.org/10.1016/0038-1101(78)90376-3">Analysis of symmetrical Van der Pauw structures with finite contacts</a>, Solid State Electronics, Vol. 21, pp. 1261-1268, 1978.
%H W. Versnel, <a href="http://dx.doi.org/10.1016/0038-1101(79)90061-3">Analysis of the Greek cross, a Van der Pauw structure with finite contacts</a>, Solid State Electronics, Vol. 22, pp. 911-914, 1979.
%H Eric. W. Weisstein, <a href="http://mathworld.wolfram.com/EllipticIntegral.html">Elliptic Integral</a>, from Wolfram MathWorld.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Van_der_Pauw_method">Van der Pauw method</a>
%F 1) Circle with contacts in the middle of each side:
%F C(d) = Pi/log(2) + (Pi^3/(64*(log(2))^2))*d^2
%F 2) Square with contacts in the middle of each side:
%F C(d) = Pi/log(2) + (Pi*K^2/(8*(log(2))^2))*d^2
%F 3) Square with complementary contacts:
%F C(d) = Pi/log(2) + (Pi*K^4/(64*(log(2))^2))*d^4
%F with K = K(sqrt(2)/2) = 1.8540746773.
%F 4) Greek cross with contacts at the cross ends:
%F C(d) = Pi/log(2) + 2*Pi/(log(2))^2*exp(Pi/2-Pi/d)
%F 5) Greek cross with contacts between the cross ends:
%F C(d) = Pi/log(2) + ((Pi/(2^12*log(2)^2)*((-3/4)!/(-1/4)!)^8))*d^4
%e 4.5323601418271938
%t RealDigits[N[Pi/Log[2], 103]][[1]] (* _Mats Granvik_, Apr 04 2012 *)
%o (PARI) Pi/log(2) \\ _Charles R Greathouse IV_, Jan 30 2016
%Y Cf. A000796 (Pi), A002162 (log(2)), A093341 (K), A131223 (2*Pi/log(2)), A259679 (log(2)/(4*Pi^2)).
%K cons,easy,nonn
%O 1,1
%A _Johannes W. Meijer_, Aug 13 2009
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