...-3.A2/-.2 i -2\ i /-2 -2\2T + Jx (<TO + a ) + (a0 - a ) J — f 'f2 where f2 — x2 + (a — <r0)2, and F and E are the complete elliptic integrals of the first and second kind with argument k, defined as f"/2 d<a f"/2 fW= ; ; — m BW- (1-fc2cos2w)1/2da> (2.4.10)...

...In the next step we perform the singularities absorbing substitution Hence, = r_ sin 2 0. where A', E are the complete elliptic integrals of the first and second kind, respectively, T/2 T/2 ^j\ - ri sin 2 0 - 2 _) = / Jl-risi r* • sn The elliptic integrals for the parameter value...

...perform the singularities-absorbing substitution £, = r-sin28. (4.17) ( Schempp, Segman Hence, where K, E are the complete elliptic integrals of the first and second kind, respectively, K(ri) = J72 I . d8, n/2 Y~r-sm e (4.19) The elliptic integrals for the parameter value r2. can be evaluated...

...J(l-msm2vfdv = - (2 - TO) E(m)-- (1 - m)K(m) , (3.159) o where m is the elliptic parameter and K(m) and E(m) are the complete elliptic integrals of the first and second kind respectively (see also note 6.2). Then we have (3.160) and we obtain Sir h cosh2 kh with m = P + q (3.161b) as is...

...= y(x) or yi — y(x')). The integrand reads F(lt x, = (yyi ^ + a - (x — f + (x — ,) M K and K' are the complete elliptic integrals of the first and second kind, respectively: 7r/2 7r/2 K(k) = f (1 - k2sin2t)-l/2dt ; K'(k) = f (1 - k2sm2t)l/2dt (32) and a2 = (y+y^ + ^-z')2,...

...[6(2 — rn)G 6 — 5(1 — m)G 4 ]/7; G 10 = [8(2 — m)G 8 — 7(1 — rn)G 6 ]/9 and K(rn), E(rn) are the complete elliptic integrals of the first and second kind, respectively. In order to find the extremum values of 8 W the Euler—Lagrange method can be used and the following...

...His explicit out-of-plane normal stress in the loop plane (ie, z = 0) az reads: <o<s<D. (99) where K and E are the complete elliptic integrals of the first and second kind, respectively, x is the distance from loop center, and R the loop radius. In order to evaluate the accuracy of the...

...tension t can be computed from the solution k of the equation 2[K(k)-E(k)]Pbot=l (12) where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively. In this formulation, for a solution to exist, it is required that 0 < k2 < 1, so that / must be less...

...(surface equation y — y(x) or j/i = y(x')). In the integrand F(x,x') = {yyi[(K-2D)/3]2 dx K and K' are the complete elliptic integrals of the first and second kind, respectively: /•T/2 K(k) = / (1 - k2sm2t)~1/2dt (11) Jo /•ff/2 K'(k) = (I- fcWi)1/2^ (12) Jo and a2 = (y + y,)2...

...y-\ (2.60) and «,/ = H0." = UQ,' = u:g = 0. In the above, 2Rr 2 i+r The quantities K(m) and £(w) are the complete elliptic integrals of the first and second kind respectively. Also, m is called the parameter of the elliptic integral and k is the modulus. From the above displacement...