Smith 



q)(x,y,z)= +4 | vsec^O e^^sec^e (y+b) ^^^^ [vs.ec'^e p cos ((9- a)] d^ . 



(21) 



- 'V ;„. 



The exponential integral in u can be evaluated with great accuracy by means of 

 a relatively simple algebraic formula, as mentioned previously. The purpose of 

 the present section is to describe a quadrature technique capable of evaluating 

 the monstrosity on the second line, known as the far-field term. 



When the substitution t = tan t^ is introduced, the far-field can be reduced 

 to an expression of the type 



2i^e"° 



;in [(/5+7t) ^1 + t^ dt] , (22) 



where a, /?, and y are measures of distance in the y-, x-, and z -directions, 

 respectively. The quantities /3 and y may vary from less than 1 to more than 

 10,000 in practical calculations, and hence under some conditions the integrand 

 in Eq. (22) is a wildly oscillating nonlinear function. It falls within the class of 

 functions covered by Eq, (19). 



Description 



The essential feature of the method of numerically evaluating Eq. 19 is re- 

 placement of the nonlinear function g(x) by a linear function plus an increment 

 S(x). If step lengths in the x-direction are chosen so that h does not vary 

 greatly over the interval of integration, then sin 5 or cos § can be approxi- 

 mated satisfactorily by low-order polynomials, and quadrature of Filon's type 

 can be performed, A detailed description of the development will now be given. 



The quantities f and g are arbitrary functions such as those sketched in 

 Fig, 15, which is drawn to illustrate specifically the five-point quadrature 

 treatment, i.e., n = 2. For simplicity, we can assume without loss of general- 

 ity that the origin of x is at the center of the range of integration. At equally 

 spaced steps of length h, the quantities f and g have values as indicated. We 

 now approximate g in the range of interest by a line segment plus an increment 

 S(x). Numerous treatments are possible, but the following appears to be as 

 simple as any: A straight line AB is passed through the two values of g at the 

 extremes of the integration range. In Fig. 15 the line passes through the points 

 (-2h, g_2) and (2h, g2). Next, we construct the line CD parallel to AB and pass- 

 ing through the origin. Then we can write 



g(x) = \x + S(x) , (23) 



where s is the difference between g and the line CD, and \ is the slope of line 

 CD. It is useful to observe that S(0) is just the value g^ and that S(2h) = S(-2h) 

 = (g2 + g-2)/2. The quantity S can have any magnitude; but d (Fig. 15) should 

 never exceed about one radian, because d measures the extent of the sine or 



352 



