Sine integral

The function Si (a ) is the sine integral [28, (5.2.1)]. The Barrett moment can be evaluated from incomplete gamma functions with complex arguments [60, (1.5.51.3)],... 

Only the sine integral remains finite at the origin, and this information is helpful in evaluating arbitrary constants of integration. It is also important to note that Ei x) is infinite at c - oo, hence Ei(—x) is also unbounded as X -oo. These boundary values are worth listing... 

Use the Laplace transform to find the value of the sine integral... 

This is called the Fourier sine integral. By similar arguments, we can derive the Fourier cosine integral as... 

This expression is similar to those of Equations 3.31 and 3.32, as must be true sinee integrals of the pressure and body forces arc equal in magnitude but opposite in sign, according to Equation 3.23. Einally, the van der Waals body force can be written as -Vq) where the van da- Waals raiergy (pv can be calculated, for instance, by integration as in Chapter 3 (Section 1). 

The plot of scattered field amplitude is shown in the upper part of Fig. 5.1 lb. It is the so-called sine-integral function. The scattering intensity is shown in the lower part of the figure. Integrating over the y and z co-ordinates we obtain the three-dimensional scattering amplitude F(q)= pFA AyA and intensity 7(q)= p"F2(A,AyA,)2. 

We have again found the scattering field amplitude in the form of sine integral. The correspondent intensity spectmm is similar to that for the parallelepiped, see Fig. 5.11b,... 

The Fourier sine integral formula can be written in the form... 

As shown by the expansions, the are unity at zero u, whereas the which include contributions from Rcross(q>0), are zero at zero u Si( ) is the sine integral Note that the explicit d ndence of Piiv(q, 9) on the scattering angle is not important for small q. For small u, with neglect of the anisotropy,... 

The sine integral function Si(jc) is given in standard tabulations (9). For large y this P(q) function asymptotically approaches (7r/2y) ( l/2y ) and, in fact, the asymptotic... 

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