By Carl M. Bender
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Additional info for Advanced mathematical methods for scientists and engineers
This integral cannot be evaluated in terms of elementary functions except when z is an integer or a half integer. Other integral representations hold for Re z S; a (see Prob. 6), rtz) does not solve a simple differential equation but it is a solution of the difference equation nz + 1) = zrtz). 3~ This difference equation for nz) can be used to evaluate rtz) [or Re z ,:; 0 from the integra! 2). nzl is a single-valued analytic function for all complex values of z except for z = 0, I. - 2, - 3, -4, , ..
It is often useful to expand a given function in terms of an orthonormal set of functions. For a discussion of the convergence of these expansions and their applications see the Rererences. In Example 5 of Sec. 1 we show how to use WKB theory to find approximate formulas for the eigenvalues E. (x). Example 7 Schriidinger eigenvaille problem. 11) has an infinite number of real positive eigenvalues Eo, E " E Physically, the eigenvalues are the allowed energy levels of a particle In the potential V(x).
B) The equation in par! (b) of the previous example may be converted to an equidimensional equation by the substitution u(y) = yl/'V(Y). The resulting equation is 3v 1 + ylu'l + 6yvv' + y'O"" + I = 0, The subsequent substitution y e' makes this equation autonomous: 30 2(t) + [v'(tj]1 + 60(t)o'(t) + v(t)v"(t) - v(t)v'(t) + 1 = O. Finally. this equation may be reduced to first order by letting w(v) v'(t): 3D' + w' + 6vw + vw dw du vw + 1 = 0. Again the final equation is very complicated. but it is the first-order equivalent of the third-order equation y" + yj 0, The reduction through two orders has followed the sequence of transformations autonomous (order 3) ....
Advanced mathematical methods for scientists and engineers by Carl M. Bender