By Xin-She Yang

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Max = exp[n ln n- n). Thus, we now can set t = n + r = n(1 + () so that r = n( varies around n and ( around 0. For n >> 1, we have n! 6: Variation of f(x). where we have used ln[n(l + ()] = Inn+ ln(l + (). The integration limits for r = n( (not () are from -oo to oo. Using (2 ln(l (3 + () = (- 2" + "3 - ... fi27r. 5 Some Special Integrals Calculus /'(0:, x) and the upper incomplete gamnla function r(a, x) so that r(x) = 'Y(O:, x) + r(a, x). 97) while the upper incomplete gamma function is defined by 1 00 r(a,x) = to:-le-tdt.

Later, Niaxwell in 1890 showed that a fluid or gaseous ring was unstable, therefore, the rings must be particulate. Suppose the whole particulate system consists of N particles (i = 1, 2, ... , N). Its total angular momentum is h. By choosing a coordinate system so that (x, y) plane coincides with the plane of the rings, and the z-axis (along k direction) is normal to this plane. If we now decompose the velocity of each particle into Vi = (vir, Vie, Viz), the total angular momentum is then N h = k · [L miri x vi] i=l N = L mi(ri i=l X Viz)· k + N N i=l i=l L mi(ri x Vir)· k + L mi(ri x Vie)· k.

This is probably the main reason why the planetary system and rings are formed. 1 Matrix 1\tiatrices are widely used in almost all engineering subjects. A matrix is a table or array of numbers or functions arranged in rows and columns. The elements or entries of a n1atrix A are often denoted as aii. 1) we say the size of A is m by n, or m x n. A is square if m = n. 3) where A is a 2 x 3 matrix, B is a 2 x 2 square matrix, and u is a 3 x 1 column matrix or column vector. 4) where (i = 1, 2, ... , m;j = 1, 2, ...