Expansion in series

Expansion in Series 60. Finite Difference Calculus 60. Interpolation 64. Roots of Equations 69. 

This equation cannot be solved by expansion in series, as the coefficients of S(p) and its first derivative result in a singularity at p = 0. Because this point is regular, the substitution Sip) = ps (p) is suggested. If the coefficient of p 2 is set equal to zero, the resulting indicial equation is... 

It is important that the degree of conversion of the initial substance be small, as the equations are derived by expansion in series of the function containing 2//(A)o- In order that only two terms might be taken, this ratio should be small. 

This integral equation is of the same form as in the normal case of control by only charge transfer and diffusion, except that l has the more complex meaning of eqn. (213) and the denominator of the right-hand side contains the two terms from the coupled chemical reactions. The equation was solved by Matsuda and Ayabe [164] by means of the method of expansion in series [75, 147]. For the present case, the result is... 

Replace the potential by its expansion in series in the vicinity of the summit of the potential curve ... 

A solution of equation (7.25) for a steady case at small velocity gradients can be easily found as an expansion in series in powers of velocity gradient. Up to the second-order terms with respect to velocity gradients, equation (7.26) immediately gives... 

The expansion in series of sin(gr,y) in the right-hand side of Equation 5.408 leads to a fairly simple and general result ... 

The stationarity coefficient, (AE), emerges formally upon the expansion in series of the time evolution operator, eas the coefficient of f in the... 

Our next step will be the derivation of the expansion over the defect encounter multiplicity. For this purpose let us present the exponential factor exp(i 5def (P — ilo)] as a finite expansion in series over Cn(Po) = exp[i5def(iln — Ho)] — 1... 

The values of the probabilities used in the transition matrix are related to the rate constants and to the time interval At. For a first-order reaction following eq. (4.1), the first term of the expansion in series of the exponential for one unit of time At = 1), leads to... 

For nonionic surfactants of the ethylene oxide type, the temperature effect is opposite to that of ionic surfactants, because of the peculiar temperature dependence of the hydration of ethylene oxide groups. As the temperature increases, the ethylene oxide chain loses its hydration water and the spontaneous curvature decreases. To a good approximation, the spontaneous curvature can be approximated by the first term of expansion in series versus temperature "... 

Consider again an oil-water-C,E system with the surfactant concentration above the CMC. Our interest now is the oil-water interfacial tension as a function of the monolayer spontaneous curvature. Experiment shows that the interfacial tension is a parabolic function of the spontaneous curvature, with a minimum at the balance point (Figure 7.6). The interfacial tension minimum as a function of the spontaneous curvature was reported for the first time by Wellman and Tartar, and, more recently, systematically studied by the Yokohama, " Hull, Paris, - and Gottingen groups. - " The dependence of the interfacial tension on the spontaneous curvature can be described by an empirical expansion-in-series - ... 

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