This paper presents tables of all-pass rational transfer functions that approximate a time delay τ to within an error ϵ for frequencies ω such that 0≤ωτ≤k. The coefficients are optimized to maximize the bandwidth k, if the order n of the transfer function and the maximum error ϵ are predetermined by the user. Two sets of tables are given, corresponding to two different error criteria: 1) ϵ = max |ϵT|, where ϵT(ω73x003C4;) represents a relative delay time error (100 ϵT is the percent error of the time delay) 2) ϵ = max |ϵø|, where ϵø(ωτ) represents a phase shift error. The tables are presented for discrete ϵ. A proof that the coefficients maximize the bandwidth and a description of the digital computer program used to obtain the tables are also given.

A procedure for solving Lagrange's equations of motion by analog computation is discussed. The problem of summing loop instabilities is minimized because the coefficient matrix of the acceleration terms in the generalized force equations is positive definite. No computer instabilities occur if external generalized forces do not contain acceleration-dependent terms (from rubbing friction). A two-dimensional example is analyzed and it is concluded that the existence of such terms will rarely cause computer instability for realistic physical systems. This paper also includes special techniques for systems which are conservative, contain cyclic co-ordinates, or are subject to nonholonomic or holonomic constraint equations.

By programming pairs of amplifiers to represent the real and imaginary parts of complex variables on an analog computer, techniques are developed which 1) permit the mechanization of the method of steepest descents, for finding roots of complex functions, in Cartesian coordinates; 2) allow the generation of complex functions utilizing the Cauchy integral theorem; 3) enable a programmer to scan the z plane efficiently when electronic mode control and track-and-hold circuitry are available. Examples are given which show that these techniques frequently provide more efficient programs, in terms of equipment, than previous methods.

By using the analog-computer independent variable time, as a parameter, a differential equation can be transformed into a set of first-order equations containing no divisions. This makes it possible, by means of other mathematical transformations to prevent unbounded functions from occurring during a computation and sometimes to continue solutions through singularities. The discussion includes a detailed application of the method to a second-order differential equation containing both zero and infinite slopes in the solution. The graph of the solution is obtained without reprogramming and with less equipment than that required by usual techniques.