sin %3F %3C%3D %3F
10 matching pages
19.23.5 , ,… ►In (19.23.8) , and in (19.23.9) . … ►With denoting any permutation of , , , …
… ►It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that . … ►
19.16.5… ►Just as the elementary function (§19.2(iv)) is the degenerate case …and is a degenerate case of , so is a degenerate case of the hyperelliptic integral, … ►(Note that is not an elliptic integral.) …
19.2.6… ►The cases with are the complete integrals: … ►
§19.2(iv) A Related Function:… ►When and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …For the special cases of and see (19.6.15). …
… ►Equations (19.25.9)–(19.25.11) correspond to three (nonzero) choices for the last variable of ; see (19.21.7). … ►then the five nontrivial permutations of that leave invariant change () into , , , , , and () into , , , , . … ►Inversions of 12 elliptic integrals of the first kind, producing the 12 Jacobian elliptic functions, are combined and simplified by using the properties of . … ►For these results and extensions to the Appell function (§16.13) and Lauricella’s function see Carlson (1963). ( and are equivalent to the -function of 3 and variables, respectively, but lack full symmetry.) …
-Homotopic Transformations… ►Lastly, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , . By composing these three steps, there result possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). … ►There are homographies that take to some permutation of , where may differ from . … ►There are automorphisms of equation (31.2.1) by compositions of -homotopic and homographic transformations. …
… ►The surface area of an ellipsoid with semiaxes , and volume is given by … ►
19.33.2 ,… ►and the electric capacity is given by ►
19.33.6… ►Let a homogeneous magnetic ellipsoid with semiaxes , volume , and susceptibility be placed in a previously uniform magnetic field parallel to the principal axis with semiaxis . …
… ►The divergence of a differentiable vector-valued function is … ►The line integral of a vector-valued function along is given by … ►If is oriented in the positive (anticlockwise) sense, then …Sufficient conditions for this result to hold are that and are continuously differentiable on , and is piecewise differentiable. … ►For a vector-valued function , …
… ►for a suitable contour . …The contour must be such that … ►for suitable contours , . …where is given by (31.10.4). The contours , must be chosen so that …
10: Bibliography S
Norm inequalities for a certain class of functions.
Israel J. Math. 10, pp. 364–372.
Numerical evaluation of integrals of the form and the tabulation of the function
Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.
Arbitrary symbols for
Comput. Phys. Comm. 1 (3), pp. 207–215.
On an identity of Ramanujan based on the hypergeometric series
J. Number Theory 69 (2), pp. 125–134.
Computation of infinite integrals involving Bessel functions of arbitrary order by the -transformation.
J. Comput. Appl. Math. 78 (1), pp. 125–130.