Cauchy Function Research Articles

Best Ordinate and Best Equations for the Position and Shape of an Absorption Maximum

Accurate values for the location of absorption maxima are important in qualitative analysis by spectrophotometry and for theoretical calculations. Accuracy should increase with sharpness of curvature. For instruments recording linearly in one of three ordinates ( T, A, or log A), experience shows that optimum peak sharpness occurs respectively at intermediate transmittance and high absorbance, but independent of log A. Using an empirical equation for the shape of a maximum, the curvature is measured by the second derivative, which can be maximized by inspection or by taking the derivative of this with respect to the ordinate, equating to zero, and solving. When nine published equations are put into the three ordinate forms and differentiated, six give results consistent with experience. On the basis of this result and other factors, four of the equations are recommended as superior, namely the Cauchy function, the Gaussian function, and two combinations of these. Comparison of numerical values for curvatures with the three ordinates predict accuracy to be about equal at T = 37%, A =0.43, and any value of log A, but greatest for A linear if A is increased.

Computer Programs for Absorption Spectrophotometry

Brief descriptions are given of twenty-two modular computer programs for performing the basic numerical computations of absorption spectrophotometry. The programs, written in Fortran IV for card input and output, are available from the National Research Council of Canada. The input and output formats are standardized to permit easy interfacing to yield more complex data processing systems. Though these programs were developed for ir spectrophotometry, they are readily modified for use with digitized visual and uv spectrophotometers. The operations covered include ordinate and abscissal unit and scale interconversions, ordinate addition and subtraction, location of band maxima and minima, smoothing and differentiation, slit function convolution and deconvolution, band profile analysis and asymmetry quantification, Fourier transformation to time correlation curves, multiple overlapping band separation in terms of Cauchy (Lorentz), Gauss, Cauchy-Gauss product, and Cauchy-Gauss sum functions and cell path length determination from fringe spacing analysis.

Cauchy Analysis of Imaginary-Dielectric-Index Bands*

A simple procedure is presented for use of the Cauchy–Lorentzian expression in calculating molecular parameters of a multiple-band system in the infrared. The Cauchy function accurately portrays the imaginary-dielectric-index spectrum for a system of damped harmonic oscillators, and thus highly accurate values of transition strength and damping constants may be calculated.

Predictive Techniques for Water Quality Inorganics

The hyperbolic relationship between quality and quantity formulated as c = K Qb, in which c is the mineral concentration, Q is the stream discharge, and K and b are regression coefficients, is taken and refined for use on rivers of the southwest or of other semi-arid regions. A modified Cauchy function is used to fit frequency curves for both quantity and quality. Gumbel's methods of extremal statistics are applied to the frequencies of extremes for quality and inorganic quality. A mathematical model for making a materials balance analysis of the chlorides in rivers of the southwest is presented.

Vibration of Stepped Beams

Summary Some results from the theory of integral equations, applicable to the problem of free lateral vibration of beams, are presented and the Cauchy function method is utilized in the solution, by successive approximations, for the lowest frequency and modal configuration of a stepped beam. The results are then com­ pared to the exact solution based on the differential equation of motion.