Series Binomial

However, in the analysis of calendering this equation is found to be difficult to work with and a useful approximation is obtained by expanding (R — using the binomial series and retaining only the first two terms. This gives... 

The left side is expanded in a binomial series, which is truncated after the quadratic term. Combination leads to... 

A simple improvement on this model can be made by remarking that the first term in brackets in Eq. (72) contains the factor 1/r2. As the amplitude of the vibration is small, a binomial series development can be made (see Section 2.10), namely,... 

We do not present the algebraic manipulations that lead to the various forms presented in Table 3.3 however, two relationships involved in generating these forms might be noted. First, the quantity (1 + zmP/mM)ul in Equation (96) may be approximated by the binomial series expansion (see Appendix A) to give... 

How does the general form of this expression compare with the Debye-Hiickel limiting law as far as agreement with experiment is concerned To see what the extra term (1 -i- Bal does to the shape of the log/. versus I curve, one can expand it in the form of a binomial series... 

Fractional distillation and fractional crystallization processes can also be interpreted in terms of the binomial series although with the... 

In order to carry out the inverse transformation with the aid of correspondence tables, the numerator has to be expanded into a binomial series, giving... 

The binomial sequences aim to improve the zero excitation profile and provide schemes that are less sensitive to spectrometer imperfections. The series may be written 1-1, 1-2-1, 1-3-3-1. .. and so on, where the numbers indieate the relative pulse widths, each separated by a delay x, and the overbar indicates phase inversion of the pulse. For off-resonance spins the pulse elements are additive at the exeitation maximum so for example, should one require 90° off-resonance excitation, 1-1 corresponds to the sequence 45x-t-45 x. Of this binomial series, it turns out that the 1-3-3-1 sequence [66] has good performance and is most tolerant of pulse imperfections by virtue of its symmetry [67]. The trajectory of spins with frequency offset l/2x from the transmitter for a net 90° pulse (1 = 11.25°) is shown in Fig. 9.26. During each... 

Consider the Binomial series discussed earlier in Seetion 2J.2... 

Inserting Equation (16-9) in Equation (16-11) and solving the binomial series for the condition, K[M < 1 gives... 

The (Newton) binomial series expansion generally looks like... 

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