Inverse sine series

The so-called Leibnitz series. We can obtain the inverse sine series... 

For the Fourier coefficients of the modulus and the phase we note that, because of the time-inversion invariance of the amplitude, the former is even in f and the latter is odd. Therefore the former is representable as a cosine series and the latter as a sine series. Formally,... 

Fig. 1. Top Scheme of an inversion recovery experiment 5rielding the longitudinal relaxation time (inversion is achieved by mean of the (re) radiofrequency (rf) pulse, schematized by a filled vertical rectangle). Free induction decays (fid represented by a damped sine function) resulting from the (x/2) read pulse are subjected to a Fourier transform and lead to a series of spectra corresponding to the different t values (evolution period). Spectra are generally displayed with a shift between two consecutive values of t. The analysis of the amplitude evaluation of each peak from — Mq to Mq provides an accurate evaluation of T. Bottom the example concerns carbon-13 Tl of irans-crotonaldehyde with the following values (from left to right) 20.5 s, 19.8 s, 23.3 s, and 19.3 s.

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

The film thickness for the ordinary shear-thinning response of lubricants which can be measured is now calculated. There are many generalized Newtonian fluid models that will describe the shear response displayed in Figures 1 and 2 [16]. The Ree-Eyring model utilizes a series in inverse hyperbolic sine to approximate power-law behavior at high shear rate. In others [16] the power-law exponent, , appears explicitly. Today the most widely used model outside of tribology is the Carreau equation [17] that was advanced to describe the results of molecular network theory. 

The infinite series expansion of the inverse hyperbolic sine is... 

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