By P F & Friedman, M D Byrd
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Extra info for Handbook of Elliptic Integrals for Engineers and Physicists
I + l + [u real] -1:;;;:snu:S:::1, -1:;;:;: cnu:;;:;: 1, k':;;:;: dnu:;;:;: 1, - oo < tn u < oo. (l Special V alues. am (- u) sn (-u) cn (- u) dn (- u) tn (- u) === = = - amK =n/2, snK = 1, cnK = 0, dnK=k', tnK = oo. 02 j am~t. snu, cnu, dnu, tn u. 01 l j amO=O, sn 0 = 0, cnO = 1, dn0=1, l tno = o. 03 ~ = cdu = sn (K-u), cn (~t + K) = - k' sd u, dn (u +K) = k'ndu, tn (u + K) = - (csu)fk'. 05 cn (u 2K) = dnu, dn (u = tnu. 06 Fundamental Relations. I l ( am(u +3K) =3n/2-j-tan-1 (k'tnu), 3K) = - cd u, sn (u ~ cn (u 3K) = k' sd u, I + + 3 K) = k' nd u, + u ( dn tn (u + 3K) = - (csu)jk'.
0 I = 0, Z(n/2, k) = 0, Z(- u 1) = - Z(u 1 ), Z (u1 ) = 0, if k = 0, Z(~t 1 ) = sn (u1 , 1) = tanhu1 , if k = 1, Z(u1 + 2K) = Z(u1 ), Z(u1 + i K') = Z(u1 ) + csu1 dnu1 - in/2K, Z(u1 +2iK') =Z(u1 ) -infK, Z(mK)=O, m=O, 1, 2, ... _ 1 - k2 sm4 ß = cos _1 [_1_- 2 si_n2 ß_±_k~ sin4ß-]. 1- k 2 sm4 ß where ' 1 The Zeta function is tabulated in j acobian Elliptic Function Tables by MILNETHOMFSON, Dover, New York, 1950. A tabulation ofKZ(ß, k) appears herein the Appendix. Byrd and Friedman, Elliptic Integrals.
Integrands Involving the Square Roots of Sums and Differences ofSquares, Va 0 (t 2 ± r~), Vt 2 ± rT. Introduction. J(tj_ d t. ~ VP ~ VP -L I ~ VP The first integral on the right side can be reduced to elementary form by the substitution t 2 = r; but the second integral is an elliptic integral and is the one we will now consider. 1 If the two factors under the radical sign involve the fourth power of the variable of integration, and if a singlepower ofthisvariable occurs in the numerator as a factor outside of the square root sign, the transformation !