Approximations for Elliptic Integrals*

Transcription

1 Approximations for Elliptic Integrals* By Yudell L. Luke Abstract. Closed-form approximations are derived for the three kinds of incomplete elliptic integrals by using the Padé approximations for the square root. An effective analytical representation of the error is presented. Approximations for the complete integrals based on trapezoidal-type integration formulae are also developed.. Approximations for the Square Root. We start with the following elementary identities: /cosh (n - l)d\ //sinh (n + l)d\ e~("+)9tanh 0 () tann0- ^ g / \ ginh fl J sinh (n -)0 ' /sinh («. - l)o\ //cosh (m - l)d\ e~ln+l)e coth 0 () coth 0 - ^ ginh e f/ ^ cogh e I r cqsh (n + )ö,,_.,,. /sinh 0 sinh nd\ /, 0 e~"* tanh 9 () tanh * = I coshfl ) / C Sh ns + coshn0 '...., a,, _., //sinh ö sinh nö\ e~~"e coth 0 () coth 0 = {cosh B0} / ( coshö j - sinhnö Now cosh (n - )6 sinh (n - )6, sinh 6 sinh n9 cosh 6 sinh 0 cosh 0 are polynomials in sinh0 of degree n. So if z = sinh0, then tanh 6 = ( - I/)-' and clearly (), () and (), () give rational approximations for ( - l/z)~ and ( - /z)', respectively. In the above remainder terms, we take e* = ( - z)' ± z', where the sign is chosen so that e* >. This is possible for all z except ^ z ^ 0. For z fixed, z ^ 0, z?, arg ( - l/z) < x, the remainder terms» 0 as n» =0. From [], we see that the approximations (l)-() are the (n, n), (n, n), (n, n) and (n -, n) approximants of the Padé matrix table. We note that the polynomials in these approximations can be expressed in terms of the Chebyshev polynomials, a fact which has also been observed by Longman []. The rational approximations can be easily decomposed into a sum of partial fractions. For our immediate applications we use the representations (), () in a slightly different form : Received February, 968. * This research was sponsored by the Atomic Energy Commission under Contract No. AT(- )

2 68 YUDELL L. LUKE (5) ( - z)-l/ = (n -f írfl + ( - z sin 0m)" + Vn(z) L m-l J (6) ( - z)l/ = - z(n + I)" - Ún\ - Wn(z), =i z eos 0 (7) m 7Wx n - ' To describe the remainder terms, we have need for the following definition. Let (8) ±( -z) where the sign is chosen so that er lies outside the unit circle which is possible for all z except z ^. Then Z ^e-(n+z/)s (9) VÁZ) = zl/(l - e-f)[l + e-("+)f] ' zl/d / _»-i-v"("+/)r (0) * «= - \_\-t+w] > and for z fixed, () lim {Vniz) and Wniz)\ = 0, z?, arg ( - «) < t. n +00. Approximations for the Incomplete Elliptic Integrals of the First and Third Kinds. We write F(<t>, k,v) = j ( - v sin ay\l - k sin a)~l/da, () v sin (f> 9^ 0, arg ( fc) < x, arg ( fc sin <^) < x, which is the incomplete elliptic integral of the third kind if v ^ 0. If ( v sin <t>) < 0, we interpret the integral in the Cauchy sense. When v = 0, we have the incomplete elliptic integral of the first kind which is usually notated as F(<j>, k). Expressions for analytic continuation of the elliptic integrals are detailed in []. Now () A i+, v) = Fit, 0,») = ( - v)'' arc tan [( - v)' tan <t>] _ /,xl/, + iv - l)l/tan K^-irin iv ) tan <j> = tan <t> if v~ if v 5, arg ( v )\ < x <j> real, if v >, Here arc tan x is given its principal value, that is, x/ < arc tan x < x/ for oo < x <». Then with the aid of (5) and with the same restrictions as in (), we get

3 APPROXIMATIONS FOR ELLIPTIC INTEGRALS 69 Fit, k, p) = Fni<f>, k, v) - Q i<t>, k, v), () Fni<l>, k, p) = (n + ír^afo, *){l -, (/c sin 0m -,)", 9Z V» sin dm arc tan jam tan <j>) \ - K ^,,!.!. s > m=i (rm(fc sin Bm v ) -I n m7r /, S \l/,<jm= ( - fc sin 0m), 7-T- / Fniir/,k,u) = (rax+)[(/x)a(x/,,)- (5) -fce (6) (/x)a(x/,,) = (-r") \-l/ = 0 if i» > sin 0, m-icrm(/c sin" 0, if kv l, arg(l -»») <x, r. r, i \ f Vnjk sin a), Q i<b,k,v) = / y da. ' o l i» sin a We shall develop a very convenient asymptotic formula for Qni<t>, k, v), but first we remark that in the application of (), (5), it might happen that for a particular set of parameters, k sin Bm = i» in which event limiting forms in these equations must be taken. This can always be avoided by a proper choice of n. Next we turn to an analysis of the error. Suppose temporarily that 0 S /! < and -x/ < </> < x/. Then from (8) (7) It readily follows that z = k sin «= ( cosh f)", zl/(l - zv' da = (fc - z)l/ Qni<t>,k,v) e-(n+/)r{sinhr/(+coghr)}rfr -*{' - f)x' vl \l/> ( + 0( + e )M_ (8) k ( - cosh -f- cosh Ü n = arc cosh fe")- \/c sin </> / Let f = ij - y. Then (8) goes into a form to which Watson's lemma is applicable. We find (i -gvn+ tan <p' Vfc sin <j>/ Qni<p,k,v) = ( - y sin <t>) (ti - ) (9) + g dr. i» sin <j> cos <fr t» sin <f> - 0(n-) n - ^] ~ Mn

4 60 YUDELL L. LUKE "» = (-h-lj ' 0 ^ t/,5 = ( - fc sin») (0) QnÍT/,k,v) = / x v//i - gyw+ \n - / \ fc / ( -,)gl/ X - Sa + yg 8g + -, n- -0(7"*) " Un, g and p as in (9) with <j> = x/. The results (9), (0) are valid for complex values of k sin 0 provided arg ( fc) < x, arg ( k sin < ) < x and ( h)/k sin ^ is replaced by ( ± 5)/k sin d> where the sign is chosen so that ( == b)/k sin <f>\ <. Clearly, for >, k and v fixed, () lim Q (<, k,v) = 0, n >oo c sin <j> ^ 0, arg ( - fc) < x, arg ( - /, sin i < X The following results illustrate the approximations and the striking realism of the formulas for the error. The column "Error" means the true error, and the column "Approximate Error, (9)", for example, means the value of Qnid>, k, v) as given by (9) with Oin~) and pn omitted. The number in parentheses following a base number indicates the power of 0 by which the base number must be multiplied. Analogous descriptors like those above are used for all other tables in this paper. 56 n F (x/, "/, -i') F (x/,, -') F.Gr/, I//, 0) Error 0.5(-) 0.89(-5) 0.500(-7) 0.56(-9) 0.6(-ll) 0.(-) Error 0.7(-) 0.7(-) 0.79(-6) 0.7(-7) 0.5(-9) 0.07(-0) Error 0.(-) 0.8(-5) 0.87(-7) 0.0(-9) 0.8(-ll) 0.(-) Approximate Error, (9) 0.5(-) 0.77(-5) 0.9(-7) 0.5(-9) 0.6(-ll) 0.7(-) Approximate Error, (9) 0.68(-) 0.(-) 0.7(-6) 0.7(-7) 0.(-9) 0.07(-0) 0.7(-) Approximate Error, (9) 0.9(-) 0.77(-5) 0.8(-7) 0.8(-9) 0.8(-ll) 0.56(-) 0.8(-7)

5 APPROXIMATIONS FOR ELLIPTIC INTEGRALS 6 n Fn(ir/,U/,-") Error Approximate Error, (0) 0.9K-) 0.98(-l) 0.855(-) 0.855(-) 0.8(-) 0.80(-) 0.8(-) 0.8(-) 0.8(-5) 0.8(-5) 0.86(-6) 0.86(-6) 0.90(-8) 0.90(-8) 0.0(-9) 0.0(-9) 0.(-) (). Approximations for the Incomplete Elliptic Integral of the Second Kind. Let Using (6) and the fact that /* j E(>, k) = / ( - fc sin a)l/da, * n [arg ( k)\ < T j arg ( /c sin <f>)\ < ir sin aüa - /i..-/,,,-, s /,,-,,, -. = a \ ( a ) arc tan [( a ) tan <f>] <f>], j a sin a a?±, arg ( a )\ <x, * > -t I /I m. we find that under the same restrictions as for (), () E(<j>, k) = En(<p, k) - Sn(<p, fc), (5) En(d>, fc) = (n + )0 - (n + l)" ± ^n 0m arc tan (pm tan 0) ^ (6) En(w/,k) = (x/)[(ti (7) Sn(<t>,k)= i Wn(k sin a)da In the above arc tan is evaluated manner of getting (9), (0) we have (8) Sni<t>, k) = X m=l Pm n m7r / 7 Nl/ - ) - (n + I)- g tan 0 / ±g (n -) I) \fcsin<7j sin0 / - = 0( âi)te'^-/'5=^-fcsin^/', Pm = ( fc COS 0m), n + tan 0, = Pn. as in the discussion following (). After the \n+ ] g - g - g tan > + 0(tT) n- " Pn, (9) t -^^^+0(n-)]-p,, 8g(n - )

6 6 YUDELL L. LUKE 5 and pn as in (8) with <p = x/. In (8), (9), we impose the same conditions as in (). Also the sign in ( ± ô)/k sin <f> is chosen as in the discussion following (0). For </> and fc fixed, (0) lim S (0, fc) = 0, arg ( - fc) < x, arg ( - fc sin <t>)\ < x. Two sample calculations /( 5 6 n (x/,'/) follow En(W,!'/) Error Approximate Error, (8) -0.00(-) -0.69(-) -0.57(-5) -0.9(-5) -0.55(-7) -0.5(-7) -0.75(-9) -0.7(-9) -0.(-ll) -0.08(-ll) -0.58(-) Error Approximate Error, (9) -0.60(-) -0.67(-) -0.8(-) -0.(-) -0.7(-) -0.8(-) -0.(-) -0.(-) -0.(-5) -0.(-5) -0.(-6) -0.(-6) -0.(-8) -0.(-8) -0.(-0) -0.5(-0) -0.86(-5) We remark that further representations for the incomplete elliptic integrals can be developed using (), () and also (l)-() together with the fact that y = yy~\ y z, but we omit such considerations.. Further Approximations for the Complete Elliptic Integrals. Consider the more general integral, () /ir/ ( - v sin i)-(l - k sin t)^dt, v <,w <, arg ( - fc) < x. Using trapezoidal-type numerical integration formulas discussed in a previous paper [], we have () B(k, v, «) = Bn(k, v, «) + Gn(k, v, u), () B(k, v, «) = Cn(k, v, «) - nn(k, v, «), () Bn(k, v, «) = ~ \JZ ( - " sin ^m)-(l - fc sin ft AU Lm=0 ( - vt\i - try ]. ßm = r n ' (5) C (fc, j», u) = -~ ^Z (. " sin <t>m) X(l k sin 0m)~", n m^l >m = (m - l)x An

7 APPROXIMATIONS FOR ELLIPTIC INTEGRALS 6 G ik, V,U>) = Z Lrnik, V, Co), (6) 00 Hn(k,v, u) = J ( )rlm(k,v, u), r=l. _. T.,. C cos tií dt (7) Ln(k,V,lc) = J-. -,. A - 'o ( v sin t) ( fc sin t) We now get an asymptotic formula for the evaluation of Ln(k, v, co). It is convenient to consider the contour integral intj, e dt In*(k,v,o>) = / (8) Mn ' ' 'c(l-vsint)(l-lpsmuy under the temporary assumptions that O á t! <, j» < and w <, as in (), where C is a rectangle with vertices at (0, 0), (x/, 0), (x/, R) and (0, R) with R > arc cosh fc-. Let R > o and so obtain.,... [*/ cos ntdt Mn(k,V,u) = / (l-f,s n,í)(l-*,bn,«r f" _e dy = ( )"sin xco / 'arc cosh k ' ( v cosh y) (fc cosh y )" Put y = arc cosh fc- - x and upon application of Watson's lemma to the resulting integral, we get )n(x/)fcv~ (l^l\ M»(fc,v,(o) = ^7 / \n/ ic\\ 7- ûj /i ç-\n (0) r a -co)/«(g +).g \ _,i i» fc ^,,5 = ( _ ky (-yu/)n» (l - «Vf cc(co-h)(g+l), n, -. fc = v, gas in (0). Notice that L (k, v, co) = M (fc, v, w). Also the series expansions for Gn(k, v, co) and Hn(k, v, co), see (6), are usually so rapidly convergent that they can be well approximated by use of the first term only. To extend the results to complex fc, we require arg ( fc) < x, and in (0), () replace ( o)/k by ( ± b)/k and choose the sign so that ( ± 8)/k\ <. Hence () lim \Gn(k, v, co) and Hn(k, v, co)} =0, v < l,co < l,fc;, arg ( - fc) < x. The foregoing results are also valid for all v, v ±, and all co, R(o>) <, x < arg co ^ x. When» >, we interpret () in the Cauchy sense. Some sample calculations based on (5) follow. The column "Approximate Error" is the value of L (fc, v, co) = Mn(fc, v, co) as given by (0), () without

8 6 YUDELL L. LUKE the 0(n~~) term. It calls for remark that for virtually the same number of terms, there is little difference in the accuracy of the formulas developed in this section and the corresponding formulas in the previous sections. However, (6) must be used with some caution because differences of large numbers occur since for m large, m ^ n, 6m is near x/ and tan dm is large. C (l'v, -'Vè) Error Approximate Error (0) 0.6(0) (-l) 0.8(-l) (-) 0.(-) (-) 0.(-) (-) 0.0(-) (-5) 0.(-5) (-7) 0.(-7) (-9) 0.86(-9) (-) n 5 6 S 0 5 C(ls/,0,- ) Error -0.08(-l) -0.09(-) -0.6(-) -0.8(-5) -0.5(-6) -0.95(-7).(-9) -0. (-) Approximate Error -0.75(-l) -0.0(-) (-) -0.(-5) -0.(-6) -0.89(-7) -0.(-9) -0.0(-ll) -0.87(-6) 5. Acknowledgement. I am indebted to Miss Rosemary Moran for the numerical calculations. Midwest Research Institute Kansas City, Missouri 60. Y. L. Luke, "The Padé table and the r-maûiod," J. Math. Phys., v. 7, 958, pp MR 0 # I. M. Longman, "The application of rational approximations to the solution of problems in theoretical seismology," Bull. Seismological Soc. America, v. 56, 966, pp P. F. Byrd & M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, New York, 95. MR IS, 70.. Y. L. Luke, "Simple formulas for the evaluation of some higher transcendental functions," /. Math. Phys., v., 956, pp MR 7, 8.