Complex Representation of Potentials and Fields for the Nonlinear Magnetic Insert of the Integrable Optics Test Accelerator

An alternative representation for the vector potential of the nonlinear magnetic insert for the Integrable Optics Test Accelerator (IOTA), first described in Sec. V.A. of the paper of Danilov and Nagaitsev, is determined from first principles using standard complex variable methods. In particular, it is shown that the coupled system consisting of the 2D Laplace equation and the Bertrand-Darboux equation is equivalent to a single ordinary differential equation in the complex plane, and a simple solution is constructed. The results are consistent with the paper of Danilov and Nagaitsev, and this concise representation provides computational advantages for particle tracking through the nonlinear insert by avoiding numerical errors caused by small denominators that appear when evaluating transverse derivatives of the vector potential near the midplane. A similar representation is provided for the spatial dependence of the two invariants of motion.