...vortex-filament of indefinitely small section, putting adydc—m^ and the corresponding part of ^ = ^,ni, we have and F and E are the complete elliptic integrals of the first and second orders for the modulus K. If we put, for sake of brevity, U=?(FE)-*F, where, therefore, U is...

...0.8 >.O 1.2 (d) where x = \u\/Ug, z' = (1 - x 2)1/2, ug is the gap frequency (2.10) and K(x) and E(x) are the complete elliptic integrals of the first and second kind, respectively (see, eg, Reference 1.41). Real parts of the functions Ip q(U) are always even and the imaginary parts...

...-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...

...[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...

...= 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,...

...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...

...(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...