## Tables of the Complex Fresnel IntegralScientific and Technical Information Division, National Aeronautics and Space Administration, 1964 - 294ÆäÀÌÁö |

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49951 ÆäÀÌÁö - Taylor's series expansions. The real and imaginary parts, accurate to five significant figures, are tabulated essentially throughout the complex plane, except in the regions xy < -6.8. Tabulation intervals in the x-direction are 0.02 for 0 ^ x ^ 10 and 0.01 for 10 = x = 20. Similar intervals apply in the ¬å and -y directions. An error analysis is presented of the methods used to evaluate the integrals. INTRODUCTION The name Fresnel integral originated in the field of physical optics where it was...

49983 ÆäÀÌÁö - Hastings, C., Jr., and Marcum, JI: Tables of Integrals Associated With the Error Function of a Complex Variable. US Air Force Project RAND Res. Memo. RM- 50, The RAND Corp., Aug. 1, 1948. 2 ¬¤ ¡Æ¡Æ 2 6. Hensman, R., and Jenkins, DP: Tables of - e / e~ dt for Complex z.

49957 ÆäÀÌÁö - Equations (7a) and (7b) indicate that the real and imaginary parts of the complex Fresnel integral evaluated at the point ¬£ are equal in amplitude but of opposite sign to the respective real and imaginary parts of the same integral evaluated at point C. Thus, with the use of equations (5) and (7) and the values of table I, the Fresnel integral and its conjugate can be evaluated for essentially all complex arguments, except in the regions xy < -6.8 where the values oscillate with amplitudes of about...

49987 ÆäÀÌÁö - Each tabulated value is listed as a decimal followed by a whole number 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| > 3, use the asymptotic series of equation (ll) in the text of this report. 20 X R(X»1Y) or 1(Y»lX) 1(X»1Y) or R(Y»1X) X R(X»1Y) or I(Y»1X) 1(X»1Y) or R(Y»1X) X R(X»1Y) or 1(Y»1X) 1(X»1Y) or R(Y»1X) X or R(X»1Y) I(Y»lX) I(X»1Y) or R(Y»1X) Y" l.12 1.U...

49975 ÆäÀÌÁö - Although this result indicates ¬Ñ¬Ý accuracy of at least eight significant figures (L ^ l) , the total effect of dropping all terms beyond the sixth term is accumulative and therefore directly proportional to the number of integration steps performed after the initial value was provided by means of the power series. Thus, the maximum accumulated fractional error becomes: Interval Final upper ¬¤ limit Initial lower limit...

49969 ÆäÀÌÁö - ... Now, by keeping the term - a sufficiently small, a high degree of accuracy can be retained with only a small number of terms of the series. This term is kept small by breaking the integration into small steps, as must be done in any case to provide a fine grid for the tabulated data. Equation...

49967 ÆäÀÌÁö - However, neither the power series nor the asymptotic series is satisfactory for moderate values of the argument. Consequently, a third method was used to evaluate equation (9); this method provided results for moderate as well as large values of the argument. The third method expresses the exponential as a Taylor>s series expansion about some point a in the complex plane before integration.

49951 ÆäÀÌÁö - ... related to those of the present report; however, the purpose of reference 3 was to demonstrate a method for computing complex Fresnel integrals and not to provide the extensive tabulated values included in the present paper. Also, the tabulations of reference k were found to be limited to a relatively small area in the complex plane. Other pertinent reports include references 5 to 8 which present tabulated values for various forms of the complex error function. These values are related to the...

49981 ÆäÀÌÁö - This procedure was decided upon after comparing results between single- and double-precision operations. For this comparison, areas were chosen where the errors due to rounding and truncation were known to be greatest. The singleand double-precision results differed only in the fifth significant figure; this result indicates that the single-precision computations are accurate to four significant figures and that as a result of increased accuracy, the doubleprecision computations are accurate to at...

49967 ÆäÀÌÁö - iy) = -x»iy n=0 du irtV1 (x » iy)2n»1 2 n n=0 (10) This result should be used only for small values of the argument for which moderate values of n can be used with good accuracy. For large values of the argument, the integral of equation (9) can be evaluated more efficiently when expressed as an asymptotic series. The integral is expanded asymptotically by repeated integration by parts, with the result that: E(x » iy) » 1)