Tables of the Complex Fresnel IntegralScientific and Technical Information Division, National Aeronautics and Space Administration, 1964 - 294페이지 |
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17 페이지 - Jp for Complex Values of p. Phil. Trans. Roy. Soc. London, ser. A, vol. 2*4-5, no. 895, Sept. 16, 1952, pp. 189-211. Hastings, C., Jr., and Marcum, JI: Tables of Integrals Associated With the Error Function of a Complex Variable.
19 페이지 - Each tabulated value is listed as a decimal followed by a whole number b which indicates the power of ten by which the decimal is to be multiplied. For computing additional or more accurate values of the integral in the region |x + iy| > J> use the asymptotic series of equation (ll) in the text of this report. Y> 1.82 FOR LARGER X VALUES, RI X...
17 페이지 - ... single-precision computations are accurate to four significant figures and that as a result of increased accuracy, the doubleprecision computations are accurate to at least the required five significant figures except for a few tabulations as explained in the section entitled "Accuracy.
6 페이지 - Fresnel integral is given by the sketches of figure 2 for limited regions of the complex plane. Since the real and imaginary parts of E(X + iy) look much alike, only the real part is included. Figure 2(a) shows R(X + iy) in the first quadrant of the xy-plane. Note the sinusoidal nature of the surface in the x- and y-directions. The frequency of these oscillations as obtained from the second derivative of the integral is Here . ply it |x + iy(.
4 페이지 - ... at point B. This symmetry allowed the number of tabulated values of the Fresnel integral to be reduced by a factor of two where double headings were used. These relationships (eqs. (5))> when combined with equations (l) and (2), also lead to the following useful equation: E*(x + iy) = -iE(y + ix...
12 페이지 - The maximum fractional error in our region of application, therefore, occurs at the points (x,y) = (l,±l) and is rtn/nl. When values are substituted for n, it can be shown that for n = 19 the maximum absolute fractional error is 0.2295 X 10...
12 페이지 - Consider the series within brackets. Since the coefficient (multipliers not containing a) of each term this series becomes smaller with each successive term for all values of n, a conservative estimate of the number of terms necessary for a desired accuracy can be obtained by ignoring these coefficients. Thus, for the purpose...
12 페이지 - Since the factor becomes smaller with each successive value of n, a 2n + 1 conservative estimate of the number of terms necessary for a desired accuracy can be obtained by ignoring its effect.
12 페이지 - Ithough this result indicates an accuracy of at least eight significant figures L ^ l), the total effect of dropping all terms beyond the sixth term is...
14 페이지 - The summation term on the right-hand side of equation (22) is the remainder after 12 terms of a Taylor expansion about zero...