Cauchy function

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

The popular radial basis function nets (RBF nets) model nonlinear relationships by linear combinations of basis functions (Zell [1994] Jagemann [1998] Zupan and Gasteiger [1993]). Functions are called to be radial when their values, starting from a central point, monotonously ascend or descend such as the Cauchy function or the modified Gauss function at Eq. (6.125) ... 

Most peaklike functions become more gaussianlike when convolved with one another. One notable exception of interest to spectroscopists is the Cauchy function, which is the familiar Lorentzian shape assumed by lines in the spectra of gases subject to pressure broadening ... 

An interesting method of fitting was presented with the introduction, some years ago, of the model 310 curve resolver by E. I. du Pont de Nemours and Company. With this equipment, the operator chose between superpositions of Gaussian and Cauchy functions electronically generated and visually superimposed on the data record. The operator had freedom to adjust the component parameters and seek a visual best match to the data. The curve resolver provided an excellent graphic demonstration of the ambiguities that can result when any method is employed to resolve curves, whether the fit is visually based or firmly rooted in rigorous least squares. The operator of the model 310 soon discovered that, when data comprise two closely spaced peaks, acceptable fits can be obtained with more than one choice of parameters. The closer the blended peaks, the wider was the choice of parameters. The part played by noise also became rapidly apparent. The noisy data trace allowed the operator additional freedom of choice, when he considered the error bar that is implicit at each data point. 

We see that the amplitude-transfer characteristic is given by 27r[l + (coRC)2] -1/2. The power-transfer characteristic is given by the square of this quantity. It has the form of a Cauchy function and attenuates high frequencies. Brodersen (1953) and Stewart (1967) have analyzed in detail the performance of other linear electrical filters applied in spectroscopy. 

In the work we have done so far with linear polymers, a Cauchy function,... 

Our discussions so far have been limited to assuming a normal, Gaussian distribution to describe the spread of observed data. Before proceeding to extend this analysis to multivariate measurements, it is worthwhile pointing out that other continuous distributions are important in spectroscopy. One distribution which is similar, but unrelated, to the Gaussian function is the Lorentzian distribution. Sometimes called the Cauchy function, the Lorentzian distribution is appropriate when describing resonance behaviour, and it is commonly encountered in emission and absorption spectroscopies. This distribution for a single variable, x, is defined by... 

The terms Lorentzian function and Cauchy function refer to the same mathematical object. 

A dispersion relationship describes the optical constant shape versus wavelength. The adjustable parameters of the dispersion relationship allow the overall optical constant shape to match the experimental results. For transparent materials, the most often used refractive index (n) wavelength (2) relationship is the Cauchy function, which has three adjustable parameters, namely Hq, A, and E ... 

Figure 8.8 shows three of the zero centered distributions in standard form and Figure 8.9 shows the corresponding probability density functions. It is seen that the Student s t function is very similar to the normal distribution function and the Cauchy function as previously mentioned is similar to the normal distribution but with a longer tail region. 

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