Symbolic evaluation of integrals occurring in accelerator orbit theory (Q1114357)

The authors discuss the equation of transerve motion along a reference curve for a particle in an accelerator \[ d^ 2x/ds^ 2+K_ x(s)x=f_ x(x,y),\quad d^ 2y/ds^ 2+K_ y(s)y=f_ y(x,y) \] where \(K_ x\) and \(K_ y\) are restorting forces which are piecewise constant function satisfying \(K_ x+K_ y=0\). They propose a method of successive approximations, starting from the solutions of the uncoupled homogeneous equations \(x=\sqrt{2J_ x\beta_ x(s)} \cos (\mu_ x(s)+\phi_ x),\) \(\mu_ x(s)=\int^{s}_{s_ 0}dt/\beta (t),\) and similar one for y. The solutions are composed by integrals of trigonometric series. They carefully use recurrence formulas for trigonometric functions of multiple arguments and represent the solution by a sum of bilinear form in trigonometric and hyperbolic functions of s. The symbolic program is implemented in REDUCE with automatic FORTRAN coding. They emphasize drastic reduction of CPU time comparing with direct numerical integration using Romberg's method.