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1 UNCLASSIFIED AD DEFENSE DOCUMENTATION CENTER FOR SCIENTIFIC AND TECHNICAL INFORMATION CAMERON STATION ALEXANDRIA VIRGINIA DOWHGRADED AT 3 TOAR INTERVALS: DECLASSIFIED ATTER 12 YEARS DOD DIR UNCLASSIFIED

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3 Office of Naval Research Contract NRORl-76 Task Order Ko.l XR-078-OU A TABULATION OF THE FRESNEL INTEGRALS Robert lurner and Auric r. Downey Marrh 15,1953 Technical Report - No. lt.'j Crufi Laboratory Harvard University Cambridge, Massachusetts,- #$?

4 Office of Naval Research Contract N5ori-76 Task Order No. 1 NR Technical Report A Tabulation of the Fresnel Integrals Robert Turner and Anne F, Downey March 15, 1953 The research reported in this document was made possible through support extended Cruft Laboratory, Harvard University, jointly by the Navy Department (Office of NavaJ Research), the Signal Corps of the U, S Army, and the U, S Air Force, under ONR Contract N5ori-76, T Technical Report No. 173 Cruft Laboratory Harvard University Cambridge, Massachusetts

5 A Tabulation cf the Presnel Integrals by Robert Turner and Anne F Downey Craft Laboratory, Harvard University Cambridge. Massachusetts Abstract A tabulation of the Fresnel integrals x I is given for 0 x by steps of 0.01, and for x g 30.0 by steps of In addition, values are given for x - n 5, n = 1-20, Differences are tabulated, to facilitate interpolation. Some applications of the integrals are listed. Alternative forms and asymptotic expansions valid for large x are given. ***** The Fresnel integrals «*> -vk / ^F «. I : i i were first obtained (although not in this form) by Augustin Jean Fresnel in 1813 in the course of the development of his theory of -X-

6 TR dlffraction of light. Since then, their chief application has been in diffraction problems. However, other applications occur. In hydrodynamics, for example, the velocity potential for surface waves generated by an impulsive pressure is readily evaluated in terms of the Fresnel integrals.* The work that initiated thi.- tabulation *<as concerned with the scattering of electromagnetic radiation by cylindrical mirrors. (The Fresnel integrals arose from the contributions to the scattered field from the singularity in the surface-current density at the edge of the mirror.) When the actual evaluation of certain integrals in the formulation was begun, it was found that they could easily be cast into forms yielding the Fresnel integrals, but that the arguments 7rere such that interpolation would frequently be required. However, it was found that no tables existed for which the tabulation was complete enough to permit satisfactory interpolation. The best previous tabulation is probably that of Watson,** which is the basis for the present work. There are two principal forms of the integrals. The first is given above, and is used principally for convenience in tabulation. The second form is that originally used by Fresnel: C(«u 2 ) = J cos( t 2 ) dt o S( 2 u2) = / sin( 2 t2) dt * o ertain other forms also exist; Cvx) :<=0- /- / J, (t) dt "Fourier Transforms," Ian N= Sneddon, McGraw-Hill, New York, 1948, pp, **"The Theory of Bessel Functions," G. N. Watson, Cambridge, London, 1922.

7 TR S(x) = \ I Ji (t) *o * wnen J, and J, are Bessel functions of order -jt and - - respectively. Further \ J H< 2) (t) dt = C(x) - is(x) (2) wnere H_i is the Hankel function of the second kind of order -j, ^P = C(x) - is(x), where 5 is the error function,. Also u / *-,r i"* dt = C(j[ u 2 O *Y ) - is( u 2 O ) I Integrals of half-order Bessel functions of order higher than may be computed by a simple integration by parts. For example, x 2 f J 3 (x) ta = G(:x) " J i (x ) This result may also be obtained from the recursion formula for the Bessel functions. More complicated integrals may be evaluated in terms of the Fresnel integrals. For example, J e-ip cos(a-o) ^ m 2 j^(p) ^2^^ CO 2^ (-i) n J n (p) V2m [cos n(a+3)c( (n-3)) + sin n(a+3) S(^(n-3))] (

8 TR173 -*- For x 2 3 "» the following asymptotic formulas give values accurate to six decimal places: C(x) = \ Vl^r [sln x(^? ~ ~^) ' cos x( "" 1? " i5 2 2TT -7)3 *** 8x-* 4x^ l6x* S(x)? -i -i/«f- [cos *( - -4r) + sin xh-p - ^^-T)]. 2 V 2TT 2X g x 3 4x 2 l6x 4 The values of S(x) at x = nn and C(x) at x = ( 2r^Z ) n were taken from Watson.

9 (X) C(X) ? UUJ ^ I OIO

10 X six) a C(X) a G >99cp ^ ^ ^2. ' ?P ? IO336O ** Q

11 S(X) C(X) A ?,224C ^ OO * ^ S no">.9q c

12 S(X) c(x) C ^ '7r\r\A I~IC. 1 W'* I \J

13 (X) C(X) i.9.5^ ^ ^.0"n ^ ^ ^ ^02 -.OI * OI ^ " ? ^ ^ _ m Aetna _ r*i > Q Cn

14 S(X) C(X) 10, ? nnqkt' OO863I OI '? ' ^

15 SCX) C(X) *2.011*69.CC A,4.2805* «. ^ -. m, *s r\r\cn *-\ * * * c.6* * * * f * Q , *

16 "\ S(X) & C(X) 12, ,577 7?i ^ ^ lL'04l , <iJ ^

17 S(X) C(X) 13, n.o CO , ^ , * , > ^ t « ^ C ^ "^.5^ ^

18 i S(X) C(X) CO ? * -.c: ^ > ^ i i ^

19 T -\ i S(X) C(X) 15, ^69, AT 7395 *\^ ft % T"f\ , ^ ^ OO OO638I ^ ^ "* ^

20 S(X) C(X) 16, ? I.5^ C O8l ^ ^ ^ ,0076^

21 S(X) C(X) C.42121? ^ C ^ ^74^ ^ H OO379O CC OO ^ OOII83 +,

22 ' \ i 1 X S(X) A C(X) A ,A OO ^ ' ! ^ ' i

23 \ I 19. Values of C(x) at x - (2n-l) Values 9f S(x) at x * ntr N C(x) N S(x) O O