Complex error function

The Voigt function can be expressed in terms of the real part of the complex error function,68-71... 

Equation [A2] contains the product of a complex exponential multiplied by a complex error function. Thus, the product of Eq. [A7] and [A 16] must be separated into its real and imaginary parts as well as the product of Eq. [A8] and [A 16]. The product of Eq. [A7] and [A 16] produces... 

G.P.M Poppe, C.M.J Wijers, More efficient computation of the complex error function, ACM Trans. Math. Softw. 16 (1990) 38. 

Schreier, F., 1992. The Voigt and complex error function a comparison of computational methods. J. Quant. Spectrosc. Radiat. Transf. 48 (5-6), 743-762. 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

Indirect evidence of isotopic fractionation among different complexes was obtained by Marechal et al. (1999) and Marechal and Albarede (2002) who observed different elution rates of Cu and Cu on anion-exchange columns (Fig. 11). These experiments were confirmed by Zhu et al. (2002) and Rouxel (2002) with similar results on fractionation coefficients. Figure 11 shows that, in HCl medium, the heavier isotope 65 is less well retained on the column than the lighter isotope 63. Marechal and Albarede (2002) used an error function approximation to the elution curve to derive the ratio of fractionation coefficients for the 63 and 65 isotopes between the resin and the eluent. From the relationship between the elution volume (position... 

An example of a deformation density and the associated error function is given in Fig. 5.16. The complex Cr(CO)6 has octahedral symmetry, but only one diagonal mirror plane is retained in the crystal. The deformation density averaged... 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

To avoid the complex form of the error function, simplified solutions have been proposed in the literature [10]. To solve for the ignition delay time (tP fig), a first-order Taylor series expansion of Equation 3.19 is conducted. The range of validity of this expansion is limited, and thus, cannot be used over a large range of incident heat fluxes. Therefore, the domain has to be divided at least into two. 

Considering that the influential quantities are independent, the absolute systematic error can be found as a full differential of the complex linear function (6.2). Then,... 

By far, singular value decomposition (SVD) is the most popular algorithm to estimate the rank of the data matrix D. As a drawback of SVD, the threshold that separates significant contributions from noise is difficult to settle. Other eigenvalue-based and error functions can be utilized in a similar way, but the arbitrariness in the selection of the significant factors still persists. For this reason, additional assays may be required, especially in the case of complex data sets. 

The error functions are more difficult to separate into real and complex parts. The procedure used by Alshawabkeh and Adrian (1997) is followed. 

In contrast to unsupervised methods, supervised machine learning techniques also take into account the biological activity for model training. In general, a regression or classification function is trained that aims to minimize an error function while keeping the complexity of the underlying function as low as possible [63, 68]. 

A large number of operations and functions commonly used in scientific disciplines are incorporated in the language by means of reserved words in the processor s vocabulary. These include the elementary mathematical and trigonometric functions some special functions such as Bessel functions, the exponential integral, the gamma, complex gamma, and error functions ... 

Standard deviation, whereas Dj and F represent the height and width of the Lorentzian spectrum, respectively. The quantity 4> is the error function. It has been assumed for simplicity that all spectral widths are identical, i.e., yj=yj+i= y and rj=Fj+ i=F similar but more complex expressions are obtained when this is not the case. 

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