36943-15-T A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS AND LEGENDRE POLYNOMIALS ELMER G. GILBERT JOSEPH OTTERMAN JOHN F. RIORDAN September 1960 COMPUTATION DEPARTMENT THE UNIVERSITY OF MICHIGAN Ann Arbor, Michigan

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A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS AND LEGENDRE POLYNOMIALS ABSTRACT This report presents tables of Fourier transforms of functions which are trigonometric functions or Legendre polynomials on the interval (-1, 1) and are zero elsewhere. This report presents tables of Fourier transforms of functions which are trigonometric functions or Legendre polynomials on the interval (-1, 1) and are zero elsewhere. These transforms are useful in the synthesis of electrical wave filters, the analysis of transient phenomena, etc. A more limited tabulation of these transforms is given in Reference 1. The terminology used in electrical engineering has been adopted: the function domain is called time and the transform domain is called frequency. The time functions whose transforms are considered as defined, for n = 0, 1,..., by a (t) = cos n7t {n(t) = sin n7Tt for It <1 n n' 6 (t) =1 d (t -1) 2nn: dtn and an(t)= On(t) = n(t) = 0 for Itl > Thus, 0 (t) coincides with the n-th degree Legendre polynomial on the interval (-1, 1). If the Fourier transform of a time function f(t) is taken as o0 F(w) = f f(t)ee dt -o0

The University of Michigan Willow Run Laboratories the transforms of the functions a (t), 3 (t), and 0 (t) are, respectively, A (c) = i e t cos n tdt n+1 2w sin w 2 2 (n7T) -wo B (w) = f e- sin n7tdt n j 27 n sin w 2 2 (n7T) - wO n(C)= J e j0 (t)dt The functions A (w) and B (c) can be evaluated from the formulas given above. The funcn n tions O can be obtained by the recursion relation 2n- 1 8=..i + Onn jw n-1 n-2 and the formulas 8 2 sin w O w 1 = 2jF[OS w sin m or from the expansion n = (_j~n~ncl y(- lk)knk)'cv w n 77 n2n+1 (1) (2n+2k+l) 2(-j) Jn (1/2)() k=0 The functions Jn(1/2)(w) are tabulated in Reference 2. Table I gives explicit expressions for 0 (t) and e (w) for n = 0, 1,...,7 and graphs of these functions. Table II gives expressions for a (t) and A (w) for n = 0, 1, 2, 3 and for 3 n(t) and B (w) for n= 1, 2, 3, 4. Table III gives values of 8 (w) for n = 0, 1,..., 7 and w = 00(100) 4500~. Table IV gives values of A (w) and B (w) for n = 1, 2, 3, 4 and 2

The University of Michigan Willow Run Laboratories o = 0~(100) 45000. The values in Tables III and IV were calculated from the expressions given in the preceding paragraph and are expected to be accurate to ten decimal places. Tables I and II and the recursion relation given above were extracted from Reference 3. 3

The University of Michigan Willow Run Laboratories TABLE 1. GRAPHS AND FORMULAS OF LEGENDRE POLYNOMIALS AND THEIR TRANSFORMS TIME DOMAIN FREQUENCY DOMAIN ~~~~~~~e 80~ 0mO, -I[t; ~)1 ='- 2 sin o ~-~~~~ ~~ ~~1 r t~~~ ~ ~ I~~/^~19~-3w t 2 sin w 2 0 co0 W. -I t e3' 655 (t3-3t) 03.j( 30:inc 30cosw 12sina. 2cos / -\2w - 3-, 3 3.-w+ - ____ 84'_______ (35t4-30t) Iinu _ s 21r' F:. J465 *3ES- 30it - 30 O'c w 12840 sin w -31 21r2 r 30 -I \ t 04 cr/ 1YL \ 2 Ga' - 20790 sin o 20790 coa + 9450aCno J -'~r 66 1231t -3m5t4+I05t'-5) v7^-a'-';5m |a 2^ -in ".V [ VI T ^ ^~~~~~ -'w^^)~ 3 + 2 + E' 847 @. i (212 8 0 sin - 21 0 cos I.1^ ^^^^,~r 84 — (35t~-30,3)) 90 sin w 2079co 2^. sin w I'? 6 5 4. -3r -2.. ~ T 3ir 4~ ~I~,J I'3"^ ~ ^' - - (6 3 ~- 7o,t + 15 t) + 8 4 0.in. 2 10cow $O.in c, + 2~o... w v > -21 in -IV \j I -7 c4 M 3 w I2 e'270270 siin 27027"0 co~ w' 811 IT ~ ~ 7c n 4- 8276,3t0 _2475 5 2in6 o 01 5nw, +E3020i ~14 w 9_ 1 I../\^ /,,~- t 87(42947693t+l3.t-31) -0 M4' "' ""'T"4.' 756 co~ M 56 sin M + c ~0 ( -

The University of Michigan Willow Run Laboratories TABLE 2. GRAPHS AND FORMULAS OF TRIGONOMETRIC FUNCTIONS AND THEIR FOURIER TRANSFORMS TIME DOMAIN FREQUENCY DOMAIN | -I I A _____________ -3 J-. o 3~'W ^-I\1 It 3 /',(t) 2=SIN't B (m) =_ —j w'.\2 SINw u I t cl(t) CSt A,(W) wS= 2' A=+ SIN: t/3 (t) =SIN r2t 4- B, (W) 2 3 - 2 w 2 3w W a2, Al A2((u) =-47rl~a. _~.__.-Il 03a(t) COS 7r t Al (W) 2 + 1S\ 2 3w -I I t A'^ A I " 7 -2 t 2 2 I /33~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I-I /33(t) =SIN 2rt AB3 S ( I) j Nw -Il\ / \ \ / I t I -IA -'L/ 3_, B lII~33 — 13 a3(t) COS 3rt A3 1.+ 2 -j. 4i, &B4 \ (t) ~SIN 4sN t B, (W) z +j \ ^ _ ^ \ _______________________________________________ — 5

TABLE 3. TABULATION OF FOURIER TRANSFORMS OF LEGENDRE POLYNOMIALS - eo el/j 6 63/j 64 65/j 6 e6 67 0 2000000000 o o 0000000000.0000000000 *0000000000 0000000000 0000000000.0000000000.0000000000 10 1.9898615401 - *1160012304 -.0040527358.0001010969.0000019611 -.0000000311 -.0000000004.0000000000 20 1.9596310721 -.2298873706 -.0161053431.0008046761.0000312479 -.0000009924 -.0000000267.0000000006 0 30 1.9098593171 - 3395892386 -.0358436968.0026928533.0001570997 -.0000074923 -.0000003022 *0000000106 40 1.8414508579 -.4431283830 -.0627531016.0063076012.0004917109 -.0000313142 - *0000016859 *0000000786 50 1.7556445434 -.5386597635 -.0961313220.0121321397.0011855340 -.0000945577 - *0000063724.0000003718 60 1.6539866863 -.6245112830 -.1351063525.0205740322.0024209223 -.0002322625 - *0000188159.0000013193 70 1.5382977489 -.6992192458 -.1786584684.0319504837.0044043188 - *0004943721 - o0000468212 *0000038361 80 1.4106331970 -.7615589153 -.2256459918.0464762716.0073572518 -.0009469193 -.0001027389.0000096375 90 1.2732395447 -.8105694691 -.2748341081.0642546674.0115064342 -.0016723440 -.0002046856.0000216457 100 1.1285065576 -.8455727830 -.3249259946.0852716242.0170732977 - o0027688871 - o0003777124.0000444853 -. 110 o9789167493 -.8661856334 -.3745954616.1093934110.0242633138 -.0043490373 - o0006548346.0000850566 120.8269933431 -.8723250647 -.4225202711.1363677808.0332554633 -.0065370461 -.0010778422.0001531666 0 130.6752479002 -.8642068394 -.4674152812.1658286564.0441922183 -.0094655618 -.0016978138.0002621947 140.5261288165 -.8423370549 -.5080645729.1973042218.0571703865 - 0132714692 -.0025752626.0004297583 150.3819718634 -.8074971790 -.5433517430.2302282099.0722331455 -.0180910528 -.0037798591.0006783449 160.2449538840 -.7607229134 -.5722875970.2639540857.0893635636 -.0240546350 - o0053896846.0010358673 170.1170506788 -.7032774450 -.5940345463.2977717429.1084798589 -.0312808612 -.0074899917.0015361025 180 oOOCOOOOOOO -.6366197724 -.6079271019.3309262606.1294326006 -.0398708268 -.0101714610.0022189689 190 -.1047295547 -.5623689173 -.6134879572.3626382072.1520039972 -.0499022537 - 0135279702.0031306030 200 -.1959631C72 -.4822649172 -.6104392726.3921249321.1759093551 -.0614239311 -.0176539056.0043231972 210 -.2728370453 -.3981275714 - 5987088960.4186222595.2008007267 -.0744506331 -.0226410718.0058545675 220 -.3348092469 -.3118139556 -.5784313876.4414059818.2262726973 -.0889587234 -.0285752729.0077874264 230 -.3816618566 -.2251757423 -.5499438525.4598125559.2518701985 -.1048826365 -.0355326561.0101883460 240 -.4134966716 -.1400173558 -.5137767159.4732584240.2770981684 -.1221124094 -.0435759255.0131264074 250 -.4307233697 -.0580559609 -.4706397081.4812574180.3014328214 -.1404924071 -.0527505457.0166715418 260 -.4340409837.0191157735 -.4214034474 4834357544.3243342401 -.1598213532 -.0630810644.0208925846 270 -.4244131816.0900632743 -.3670771204.4795441957.3452599519 -.1798537403 -.0745676879.0258550726 280 -.4030380563.1535392679 -.3087828559.4694670255.3636791237 -.2003026529 -.0871832442.0316188300 290 -.3713132497.2085079432 -.2477274724.4532275749.3790869773 -.2208439933 -.1008706649 o0382353995 300 -.3307973373.2541635694 -.1851723389.4309901278.3910190188 -.2411220569 -.1155411087.0457453851 310 -.2831684743.2899432025 -.1224021342.4030581331.3990646705 -.2607563584 -.1310728359.0541757835 320 - 2301813572.3155332729 -.0606933120.3698687513.4028799035 -.2793495709 -.1473109266.0635373879 330 -.1736235743.3308699976 -.0012830795.3319838618.4021984934 -.2964963981 -.1640679150.0738223537 340 -.1152724160.3361337216.0546603217.2900777549.3968415499 -.3117931672 -.1811253859.0850020159 350 -.0568531869.3317374383.1060652856.2449218209.3867250186 -.3248479008 -.1982365547.0970250510 360.0000000000.3183098862.1519817755.1973666321.3718649011 -.3352906025 -.2151298233.1098160696 370.0537800416.2966737469.1916029086.1483218820.3523800030 -.3427834764 -.2315132709.1232747214 380.1031384775 2678195867.2242826774.0987347073.3284920831 -.3470307918 -.2470800144.1372753828 390.1469122552.2328762795.2495493933.0495669587.3005233492 -.3477881079 -.2615143385.1516674881 0 400.1841450858.1930787270.2671145571.0017720174.2688913166 -.3448705784 -.2744984704.1662765463 410.2141029927.1497337474.2768769814 -.0437282340.2341011210 -.3381600776 -.2857198491.1809058716 420.2362838123.1041850340.2789221176 -.0860656820 91967354452 -.3276109112 -.2948787163.1953390338 430.2504205638.0577780931.2735166630 -.1244469117.1574422916 -.3132539103 -.3016958404 92093430166 440.2564787631 00118260562.2610986479 -.1581722456.1169208871 -.2951987463 -.3059201699.2226720489 450.2546479089 -.0324227788.2422633197 -.1866523982.0759060695 -.2736343467 -.3073362086.2350720519 c 460.2453275125 -.0738148836.2177452432 -.2094223124.0351515422 -.2488273452 -.3057709008.2462856276 470.2291081754 -.1113181977.1883971355 -.2261518226 - 0045875767 -.2211185480 -.3010998220.2560574886 480.2067483358 -.1440449720.1551660298 -.2366528744.0425724627 -.1909174537 -.2932524787.2641402165 490.1791474021 -.1712702538.1190674264 -.2408831212 - 0780980908 - 1586949194 -.2822165402.2703002166 500.1473160686 -.1924456320 o0811581336 -.2389458173 - 1105100825 -.1249741136 -.2680408461.2743237261 I' 0 0 0 (I,

(u CD C TABLE 3. (Continued) CD -I w 62 6e/j e2 e3/j e4 e 5 e 6 e 7/j 500.1473160686 - 1924456320.0811581336 -.2389458173 -.1105100825 - 91249741136 - *2680408461.2743237261 510 *1123446657 -.2072079929.0425085276 - 2310860245 - *1392204476 -.0903199527 -.2508370614 *2760227229 0 520.0753704259 - 2153831725.0041749637 -.2176832476 -.1637218045 -.0553272558 - 2307798829.2752405763 530 *0375445574 - 2169845291 -.0328269393 -.1992407077 -.1835996970 - *0206078967 - *2081057358.2718572818 540.0000000000 - 2122065908 -.0675474558 -.1763715525 - 1985426763.0132227376 -.1831099400.2657941222 550 -.0361793007 - 2014140628 -.0991257138 -.1497823806 - *2083498734.0455596479 -.1561423626 *2570176096 560 - *0699868240 - 1851265943 -.1268098911 - 1202545267 - 2129358517.0758221349 -.1276016168.2455425694 570 -.1005189114 - 1639998102 -.1499741586 -.0886236143 -.2123326033.1034673649 -.0979279075 02314342515 580 -.1269966109 - 1388032021 -.1681319781 -.0557579198 -.2066886235.1280032392 -.0675946596 *2148093655 590 -.1487834356 -.1103955447 -.1809454634 -.0225361263 -.1962650811 *1490002251 -.0370991074.1958359702 600 -.1653986686 -.0796985564 -.1882306232.0101749459 -.1814291729.1661018342 - *0069520482.1747321664 ( 610 -.1765259712 -.0476695546 -.1899584185.0415420492 -.1626448306 *1790334609.0223330034.1517635775 620 - 1820171867 -.0152738690 -.1862516782.0707860913 - 1404610169.1876093350.0502510057.1272396301 630 -.1818913635.0165422341 -.1773780293.0972010631 -.1154979106.1917373891.0763166527 *1015086790 640 -.1763291496.0468773929 -.1637391023.1201708265 -.0884313393 1914218932.1000754302.0749520539 650 -.1656628345.0748990810 -.1458563673.1391833446 -.0599758633.1867637670.1211141855.0479771336 660 -.1503624260.0998650181 -.1243540440.1538420020 -.0308669500.1779585379.1390709358.0210095837 670 -.1310182493.1211415329 -.0999395990.1638737510 -.0018427033.1652919780.1536436549 -.0055150797 680 -.1083206387.1382184448 - 0733824025 1691339003.0263743778.1491335109.1645978018 -.0311604181 690 -.0830373616.1507201327 -.0454911551.1696074602 90530951369.1299275394.1717723777 -.0554982861 700 -.0559894592.1584125745 -.0170907223.1654070477.0776805351.1081829031.1750843370 -.0781181204 710 -.0280262189.1612062533.0110009837.1567674509.0995570303.0844607200.1745312144 -.0986360768 720.0000000000.1591549431.0379954439.1440370363.1182301660.0593609173.1701918748 -.1167037906 730.0272583773.1524505011.0631546935.1276662723.1332960225.0335077852.1622253395 -.1320165363 740.0529630019.1414138999.0858106491.1081937105.1444502449 0075349208.1508676918 -.1443205740 750.0763943727.1264828347 91053821031.0862298364.1514944229 -.0179300562.1364271136 -.1534194851 760.0969184662.1081963262.1213889643.0624392517.1543396738 -.0422806263.1192771529 -.1591793196 770.1140028922.0871768178.1334634003.0375216909.1530073505 -.0649459439.0998483722 -.1615324038 780.1272297451.0641103232.1413576333.0121924014.1476268795 -.0854044886.0786185664 -.1604796894 790.1363048638.0397252253.1449482350 -.0128375729.1384308031 -.1031964424 o0561017819 -.1560915554 800.1410633197.0147703532.1442368656 - 0368806696.1257471818 -.1179344762.0328364003 -.1485070174 810.1414710605 - 0100070305.1393475027 - 0592911280.1099895771 -.1293126675.0093725759 -.1379313311 820.1376227509 -.0338827051.1305203021 -.0794818664.0916449016 -.1371133233 -.0137406620 -.1246320243 830.1297359547 - 0561758363.1181023222 - 0969394785.0712594777 - 1412115338 -.0359683222 -.1089334257 840.1181419062 -.0762676335.1025354293 -.1112369629.0494236918 -.1415773445 -.0568020552 -.0912098036 850.1032732083 -.0936178274 90843417739 - 1220438726.0267556660 - 1382754969 - 0757721295 - 0718772593 860.0856488771 -.1077785421 90641072879 - 1291336411.0038843944 -.1314627511 -.0924584764 -.0513845579 870.0658572178 -.1184052205 90424637026 -.1323879270 -.0185672012 -.1213828691 -.1065005198 -.0302031060 880.0445370698 -.1252643506.0200696200 -.1317978991 -.0399988386 -.1083593971 -.1176055401 -.0088163127 0 890.0223579948 - 1282378365 - 0024088146 - 1274624718 - 0598486256 -.0927864461 - 1255553552 o0122914138 900.0000000000 -.1273239545 - o0243170841 -.1195835862 -.0776075769 -.0751177201 -.1302111453.0326457665 910 - 0218666103 -.1226349355 - 0450307781 -.1084587073 -.0928325179 -.0558540882 -.1315162968.0517934652 920 -.0426006755 - 91143913082 - 90639728684 -.0944707899 -.1051570454 -.0355300355 - 1294971888.0693127372 930 -.0616083651 - 91029132278 - 0806293122 -.0780760273 - 91143002670 - 90146993542 -.1242619019.0848230899 940 - 0783596110 -.0886090979 - 0945625706 - 0597897564 - 1200731077.0060795422 -.1159968809.0979941297 950 -.0924023442 -.0719618631 -.1054226949 -.0401709395 -.1223820392.0262582910 -.1049616373.1085532013 C 960 - 1033741679 -.0535134125 -.1129557075 - o0198056765 -.1212301572.0453128519 -.0914816292.1162916502 970 - 91110111778 - 90338475768 - s117009C849.0007097775 -.1167156098.0627570008 - 90759395025.1210695461 980 -.1151537304 -.0135722374 -.1175342362.0207860059 -.1090274476 0781546930 -.0587649224.1228187405 990 -.1157490495.0066989212 -.1145859589.0398569608 -.0964390402 0911309748 -.0404232598.1215441752 1000 - 1128506558.0263644817 -.1083189352.0573955708 - 0852992674.1013811630 -.0214034270.1173234014 0 0 0 \I (>' tfl

00 -I 0) C TABLE 3. TABULATION OF FOURIER TRANSFORMS OF LEGENDRE POLYNOMIALS (D -- (/) Ho e /i 92 e 3/i 84 e85/ 86 87/j 1000 -,1128506558.0263644817 -,1083189352,0573955708 -,0852992674.1013811630 -.0214034270.1173234014 1010 -.1066146955.0448526669 -.0989814227.0729280023 -.0700217522.1086780596 -.0022051823 *1103043164 1020 -.0972933345.0616375264 -.0869063635.0860461923 -.0530724560.1128770253.0166737682.1007011674 1030 - *0852254631.0762534890 -.0725002114.0964183246 -.0349559964.1139187902.0347505325 *0887889212 1040 -.0708250330.0883078870 -.0562298333.1037969838 -.0162010798.1118299467.0515693276 *0748961364 1050 -.0545674091.0974911271 -.0386078944.1080247909.0026545362.1067211304.0667129876.0593965169 1060 -.0369741712.1035842513 -.0201771700.1090373960.0210791302.0987829557.0798133858.0426993594 1C70 -.0185968368.1064637159 -.0014942529.1068637831.0385617343.0882798377.0905605037.0252391363 n 1080.0000000000.1061032954.0168868639.1016239156.0546260771.0755418850.0987099133.0074644789 1090.0182556105.1025731067.0344308566.0935238280.0688433744.0609551002.1040884843 -,0101731576 1100.0356296559.0960358305.0506363316.0828483392.0808436236.0449501702.1065981740 -.0272308582 1110.0516178194.0867402865.0650498527.0699516287.0903251065 *0279901635.1062178097 -.0432853488 1120,0657661021.0750125955.0772783480.0552459741.0970618552.0105574814.1030028287 -.0579435505 1130.0776833867.0612452249.0869995626.0391889959 *1009088975 -.0068595754.0970829932 -.0708523143 1140.0870519309.0458842730.0939702866.0222697958.1018051613 -.0237802522.0886581535 -.0817070892 1150.0936355151.0294153927.0980321530.0049943987.0997739841 -.0397444307.0779921846 -.0902592995 1160.0972850481.0123487868.0991148757 -.0121290755.0949212431 -.0543250062.0654052696 -.0963222447 1170.0979415034 -.0047962690.0972368727 -.0286050828.0874311859 -.0671392209.0512647466 -.0997753722 1180.0956361489 -.0215069179.0925032957 -.0439647502.0775601071 -.0778586495.0359747748 - 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*0650275743 -.0652580392 -.0506467658 - *0808833634 - *0185633210 -.0898830553 *0329358762 1310 -.0670093336 -.0532967041 -.0740024852 -.0371133832 -.0853651429 -.0035106643 -.0870541572.0459869428 1320 -.0751812130 -.0401425858 -.0804084874 -.0226915746 - *0873031239.0114136564 -.0818535045.0576016749 1330 -.0809630394 -.0259802865 -.0843206953 -.0078178054 -.0866782071 *0257887051 -.0744575879 *0674874811 1340 -.0842169073 -.0112487627 -.0856598326.0070644719 -.0835453915.0392146100 -.0651012512.0754014384 1350 -.0848826363 *0036025310 -.0844239478.0215178491 -.0780312267.0513235822 -.0540705814.0811563299 1360 -.0829784234.0181271536 -.0806873732.0351236460 -.0703292433.06i7899102 -.0416944552.0846251397 1370 -.0785991551 *0318949061 -.0745974539.0474938815 -.0606935176 o0703386564 -.0263349850.0857438942 1380 -.0719124646.0445043915 -.0663691737.0582821942 -.0494305750 *0767528246 -.0143771388.0845127824 1390 -.0631526813.0555945442 -.0562778624.0671934075 -.0368898836.0808788116 -.0002178259 *0809955358 0 1400 -.0526128817.0648548022 -.0446502250.0739914788 -.0234532277.0826300064.0137452415.0753170948 1410 -.0406353046.0720336402 - *0318539779.0785056206 -.0095232792.0819884528.0271246041.0676596377 1420 -.0276004376 *0769452331 - *0182864156.0806344389.0044882966.0790045487 *0395537185.0582570876 1430 -.0139151157.0794740791 -.0043622570.0803479914.0181728519.0737948100.0506969377.0473882594 1440.0000000000.0795774715.0094988610.0776877332.0311365378.0665377819.0602585343.0353688403 1450.0137231830.0772857761.0228848702.0727643744.0430115191.0574682319.0679905250.0225424308 C 1460.0268442613.0727005339.0354033717.0657537402.0534663050.0468698069.0736990881.0092708979 1470.0389767208.0659904759.0466929978.0568907762.0622149091.0350663780.0772494057 -.0040756927 1480.0497689421.0573855924.0564337102.0464618984.0690245850.0224123331.0785688101 -.0171292857 1490.0589142463.0471694608 o0643557450 o0347959287 *0737219323.0092821031.0776481596 -.0295339062 1500 *0661594675.0356700808 *0702469576.0222539002.0761972122 -.0039407718.0745414219 -,0409553154 "" 00 0

CD c TABLE 3. (Continued) CD C,) eo elj e2 3 /j 94 9 5/j 9 e7 /j 1500 0661594675 0356700808 *0702469576.0222539002 *0761972122 - *0039407118.0745414219 -.0409553154 1510.0713118162 90232495069 e0739583697.0092180418.0764067699 -.0168747191.0693634904 -.0510899895 0 1520 s0742438525 e0102926026 *0754077788 -.0039197246.0743735119 - *0291510371.0622863110 - 0596731790 1530.0748964438 -.0028047386.0745813464 -.0167694327 e0701854483 -.0404243149.0535334425 -.0664858339 1540 *0732796466 -.0156475519.0715331427 -.0289545232.0639923610 -.0503820750 *0433732162 -.0713602108 1550.0694715112 -.0278535780.0663826871 -.0401227644.0560007159 -.0587533722.0321107011 -.0741840170 1560 *0636148725 -.0390645149.0593105806 -.0499563288.0464669859 -.0653161106 *0200787089 -.0749029879 ~~ 1570.0559122465 -.0489564141.0505523817 - *0581807393.0356895947 -.0699028847.0076281023 -.0735218369 fl 1580.0466190091 -.0572489237.0403909299 -.0645724358.0239997297 -.0724051883 -.0048823138 -.0701035632 1590.0360350815 -.0637131190.0291473592 -.0689647563.0117513049 - 90727758894 -.0170960313 -.0647671500 ~. 1600.0244953884 -.0681777030.0171710860 -.0712521741 -.0006896278 -.0710299148 -.0286689139 -.0576837267 (Q 1610.0123593884 -.0705334140.0048290774 -.0713926865 -.0129557036 -.0672431435 -.0392788288 -.0490713147 1620.0000000000 -.0707355303 -.0075052729 -.0694083066 -.0246889897 -.0615495571 -.0486345628 - e0391883118 1630 -.0122077395 -.0688044195 -.0194633276 -.0653836633 -.0355513372 -.0541367466 -.0564837777 -.0283259080 1640 -.0238979399 -.0648241411 -.0306921162 -.0594627730 -.0452340439 -.0452399204 -.0626197888 -.0167996491 1650 -.0347247149 -.0589391646 -.0408646519 -.0518440977 -.0534665525 -.0351346006 -.0668869813 -.0049403930 1660 -.0443723098 -.0513493259 -.0496893574 -.0427740534 -.0600239412 -.0241282302 -.0691847263.0069150871 1670 -.0525642078 -.0423031899 -.0569183292 -.0325391778 -.0647330016 -.0125509442 -.0694696946 0184335851 1680 -.0590709531 -.0320900376 -.0623542097 -.0214572012 -.0674767392 -.0007457778 -.0677565184.0292947657 1690 - 90637164748 -.0210307316 -.0658554786 - 90098672967 -.0681971835.0109413998 -.0641167951.0392000287 1700 -.0663827387 -.0094677439 -.0673400242.0018801948 -.0668964414.0221719324 -.0586764769.0478806902 1710 -.0670126076 90022453448 - 0667869080.0134342569 - 90636359819.0326241158 -.0516117330.0551052551 1720 -.0656108464.0137545581 -.0642362915.0244535942 -.0585341902.0420023491 - *0431434166.0606855839 1730 -.0622432615.0247161258 -.0597875453.0346166287 -.0517622811.0500454392 -.0335303059 s0644817891 1740 -.0570340237.0348066600 -.0535956190.0436308059 -.0435387068 0565338273 - 0230613236.0664057323 1750 -.0501612726.0437326077 - 90458657974.0512409409 -.0341222388.0612955366 -.0120469639 90664230351 1760 -.0418511559.0512387055 - 0368470168.0572363718 -.0238039385.0642106826 -.0008101816.0645535575 1770 -.0323704969.0571151944 - 90268239552.0614567182 -.0128982637.0652144270.0103229938.0608703413 1780 -.0220183267.0612035953 -.0161081451.0637960917 -.0017335790.0642983057.0210328462.0554970623 1790 -.0111165449.0634008865 -.0050283831.0642056494.0093576443.0615099055.0310150956.0486040714 1800.0000000000.0636619772.0060792710.0626944312.0200486511.0569509157.0399894946 90404031488 1810.0109937102.0620004213.0168816038.0593284748.0300279565.0507736258.0477076577 90311411304 1820.0215344074.0584873691.0270581646.0542282520.0390083568.0431759839 90539599053 0210925922 1830.0313091691.0532488076.0363106969.0475645189.0467351359.0343953744.0585809324 90105518102 1840.0400315404.0464611895.0443718068.0395527187.0529932345.0247013067.0614541513 -.0001757724 1850.0474498524.0383455989.0510126216.0304461181.0576131829.0143872370.0625145924 -.0107823252 1860.0533544092.0291606436 90560492186.0205278918.0604756331.0037617698.0617502938 -.0209664054 1870.0575833382.0191942990.0593476467 90101024015.0615143712 -.0068604994.0592021497 - 90304414797 1880.0600269446.0087549600.0608274056 -.0005140656.0607177372 -.0171682309.0549622378 -.0389439518 0 1890.0606304545 -.0018380260.0604632940 -.0110028185 90581284234 -.0268624058.0491706841 - e0462404718 1900.0593950820 -.0122640542.0582855895 - 90210522602.0538416740 -.0356649476.0420111683 -.0521343236 1910.0563774044 -.0222108946.0543785711 -.0303670795.0480019541 -.0433266525.0337052119 - e0564707170 1920.0516870839 -.0313839746.0488774444 -.0386768686.0407982000 -.0496342030.0245054219 -.0591408406 1930.0454830192 -.0395150359.0419637786 -.0457439152.0324578030 -.0544160686.0146878953 -.0600845698 1940.0379680589 -.0463699157.0338596046 -.0513699478.0232395178 -.0575471316.0045440098 -.0592917623 1950.0293824510 -.0517552265.0248203648 -.0554016422 90134255125 - 90589519124 -.0056281559 -.0568021152 C 1960.0199962354 -.0555237461.0151269350 -.0577347398.0033128031 - 90586063146 -.0155325231 - 90527035982 1970 *0101008200 -.0575783706.0050769657 -.0583166668 -.0067956699 - 90565378502 -.0248835796 -.0471295186 1980.0000000000 -.0578745248 -.0050241909 -.0571475931 -.0166000552 -.0528243517 -.0334145686 -.0402543119 1990 -.0099993042 -.0564209738 - 90148726967 -.0542799116 - 90258124298 -.0475912234 -.0408850623 - 90322881883 2000 -.0195963107 - 90532790283 -.0241753059 - 00498161708 -.0341652031 - 0410073220 -.0470877087 - 90234707967 0~ 0 0" ^0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~CD C,)

-I-I (D TABLE 3. TABULATION OF FOURIER TRANSFORMS OF LEGENDRE POLYNOMIALS'980 i / i ~82 83/ij4 es5/ 86 97/j 2000 -,0195963107 -.0532790283 - *0241753059 -.0498161708 - *0341652031 - *0410073220 -.0470877087 -.0234707967 2010 - *0285053629 -.0485601812 -.0326580397 - *0439055348 - *0414188421 - *0332796027 - *0518539608 -.0140640950 2020 - o0364643734 -.0424222635 -.0400742001 -.0367388909 -.0473686971 - *0246466940 -.0550586312 -.0043446378 2030 -.0432424764 -.0350642477 -.0462114913 -.0285427615 -.0518507318 -.0153716024 -.0566231493.0054044947 2040 - *0486466672 - *0267198680 -.0508980432 -.0195722135 - *0547460023 - *0057337655 - *0565174367.0149018903 2050 -.0525272402 -.0176502596 -.0540071700 - *0101029843 -.0559837596.0039793076 - *0547603582 *0238758863 2060 - *0547818717 -.0081358473 -.0554607305 -.0004230662 -.0555430992 *0134805583 -.0514187448.0320723067 2070 - *0553582411 *0015322674 -.0552310056.0091759957 - 90534531212.0224918142 -.0466050312.0392616358 2080 - *0542551230 *0110611562 -.0533410495.0184078317 - *0497916084.0307518900 -.0404735851 *0452454250 2090 - 0515219342 *0201648856 - *0498635186.0269997400 - *0446822755 *0380241191 - *0332158463 *0498617562 2100 -.0472567625.0285730441 -.0449180270 *0347006950 -.0382906824.0441030999 -.0250544270.0529896085 ~ 2110 -.0416029512.0360387241 - *0386671216.0412886361 -.0308189453.0488204652 -.0162363516.0545520138 -. 2120 - *0347443558.0423457226 -.0313110098.0465768281 - *0224994109.0520495128 -.0070256413.0545179160 ( 2130 -.0268994270.0473147534 -.0230812077.0504191097 -.0135874916.0537085714.0023045362 *0529026923 Q 2140 -.0183143091.0508084930 -.0142333107.0527138870 -.0043538795.0537630133.0114799202.0497673298 2150 -.0092551700 *0527353200 -.0050391077 *0534067609 *0049236244.0522258651.0202331770 *0452162914 2160.0000000000.0530516477.0042217160.0524917252.0139684198 90491570106 *0283116520 *0393941420 2170.0091698688.0517627931.0132700388.0500109101.0225133100.0446610212 *0354846318 *0324810418 2180 *0179782667.0489223686 *0218356684.0460528924 *0303083540 *0388836874 *0415499058 *0246872446 2190.0261624564.0446302296.0296653657.0407496354 *0371281396.0320073654.0463394440.0162467676 2200.0334809247.0390290506.0365302882.0342721612.0427782660.0242452853.0497240287.0074104214 2210.0397204647.0322996468.0422326368.0268250951.0471008509.0158349957.0516167163 -.0015615926 2220.0447023429.0246551910.0466113137.0186402550.0499789063.0070311453.0519750344 -.0104073013 2230.0482873733.0163345106.0495464296.0099694853 *0513394637 -.0019021790.0508018605 -.0188705729 2240.0503797570.0075946732 *0509625375 *0010769537 *0511553653 - 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*0247788327 *0433533580 2470 -.0355393632.0306454615 -.0334067450.0345200880 -.0278014869.0403242057 -.0175122259.0456051354 C 2480 -.0297008203.0360822414 -.0271999814.0392242659 -.0208565563.0435609328 -.0097862202.0465001338 2490 -.0230103532.0403845772 -.0202225631.0427112188 -.0133429614.0454744551 -.0018327454.0460226927 2500 -.0156770486.0434316285 -.0126909098.0448858996 -.0054899465.0460182823.0061113283.0441974851 00 Q 0 (/>

C TABLE 3. (Continued) 190 el/J 62 3/J e4 5/i e6 87/J 2500 -.0156770486.0434316285 -.0126909098.0448858996 - *0054899465.0460182823.0061113283 *0441974851 2510 - *0079277352 *0451413871 -.0048364074.0456933905.0024648950 *0451869950 *0138112158 *0410885022 2520.0000000000.0454728409 *0031016689.0451202366.0102827776 *0430160947.0210411297 *0367968950 2530 *0078650654.0444268857.0108834129.0431945267.0177308629 *0395806404.0275908787 *0314577424 2540.0154301659.0420459681.0182755083.0399847290.0245891671.0349927213.0332719576 *o252358490 2550.0224689331.0384124808.0250581955.0355973262.0306570331.0293978488.0379229544.0183207005 2560.0287726697.0336459666.0310317773.0301733301.0357589771.0229703824.0414141199.0109207318 _. 2570 *0341565086.0278992259 *0360224711.0238837871.0397497401 *0159081329 *0436509688 *0032570778 C 2580.0384648067.0213534529 *0398874377.0169244183 *0425184006.0084263082.0445768157 -.0044430035 2590 *0415756148.0142125537.0425188419.0095095611 *0439914303.0007509860 o0441741761 - *0119528442 _. 2600.0434040984.0066968245.0438468289.0018655971.0441346119 -.0068876886 *0424650002 -.0190530151 e 2610.0439048119 -.0009638163.0438413376 -.0057759305 *0429537686 -.0142623776.0395097401 -.0255377177 2620.0430727694 - *0085368333 *0425127031 - *0131853037.0404942925 -.0211552917.0354052856 -.0312207324 Q 2630.0409432861 -.0157941061.0399110401 -.0201415094 *0368394948 - *0273645999.0302818381 - *0359407519 2640.0375906065 -.0225187737.0361244353 -.0264388072 *0321078351 - *0327103189.0242988217 - *0395659474 2650.0331253687 -.0285116899.0312760108 -.0318927906.0264491217 -.0370395104.0176399580 -.0419976375 2660.0276909903 -.0335972947.0255199567 -.0363457643 *0200398017 -.0402306382.0105076575 -.0431729605 2670.0214590935 -.0376287328 *0190366580 -.0396712799 *0130774900 -.0421969597.0031168965 - *0430664747 2680.0146241125 -.0404920680.0120270700 -.0417777030 *0057749049 -.0428888594 -.0043112357 - *0416906494 2690.0073972548 -.0421094748.0047065167 -.0426107082 -.0016466046 -.0422950608 -.0115561262 -.0390952376 2700.0000000000 -.0424413182 -.0027018982 -.0421546379 -.0089637426 -.0404426891 -.0184041683 - 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*0347728639.0249249842 - *0291824510.0326603602 2820 -.0351912061 *0210326550 -.0339092036.0244774320 -.0304279359.0300414500 -.0237138468 *0363049761 2830 -.0310184548.0266555692 -.0293994601.0296316558 -.0252000320 *0342234223 -.0175783238.0388499696 2840 -.0259359276 *0314324811 -.0240335172 90338568105 - *0192521910 *0373524526 -.0109629245 *0402276924 2850 -.0201037823.0352249344 -.0179793190.0370321960 -.0127679086.0393423453 -.0040676792.0404054299 2860 -.0137037138 *0379251774 -.0114243929 *0390695297 - *0059454998 *0401415116.0029004202 *0393861397 _. 2870 -.0069333155.0394592005 -.0045700622.0399153770 *0010079443 *0397342764 *0097336132.0372081320 2880.0000000000.0397887358.0023747152.0395525185.0078828217.0381411047.0162295466.0339437093 2890.0068853340.0389121644 *0091996968.0380002201 *0144733181 *0354177494.0221972500 *0296968104 0 2900.0135146970.0368643129.0156997000.0353134050.0205835495.0316533517.0274627350.0245997363 2910.0196892713.0337151509.0216807516.0315807599.0260333666.0269675577.0318740538 o0188090581 2920.0252253542.0295674386 *0269658571 *0269218396 *0306636497.0215067439.0353056710.0125008339 2930.0299598044.0245534007.0314002204.0214832640 *0343409380.0154394697.0376620236.0058652803 2940.0337548303.0188305341.0348557590.0154341261.0369612597 *0089512970 90388801668 -.0008989422 2950.0365019805.0125766820 *0372347847.0089607566 *0384530541.0022391365 *0389314350 -.0075906567 C 2960.0381252215.0059845274.0384727433.0022610041 *0387791021 -.0044947088.0378220729 -.0140121368 2970.0385830165 -.0007443246 *0385399391 -.0044617923 *0379374161 - 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-H C TABLE 3. TABULATION OF FOURIER TRANSFORMS OF LEGENDRE POLYNOMIALS 00 0 /2 83 / 04 85/ij 6 97/ | 3000.0330797337 -.0197303695.0319492668 -.0227812998 90289036314 - *0277494681 *0230738976 - *0334782948 3010.0291635306 - *0250262397.0277343964 -.0276658807 *0240480285 - *0317856996.0173925299 -.0360896003 0 3020 *0243900776 -.0295296939.0227093556 -.0316839161 *0185015798 -.0346430426.0112300725 -.0376127992 3030 *0189094982 -.0331097807.0170312302 -.0347200408.0124354603 -.0368363756 *0047733292 -.0380097720 3040 *0128923097 -.0356643149.0108757820 -.0366892103 *0060353352 - *0377129581 -.0017833249 -.0372760170 3050.0065241362 -.0371227742 *0044320264 -.0375390634 - *0005043045 -.0374538008 -.0082437772 - *0354405755 3060 *0000000000 -.0374482219 -.0051035540 -.0372512860 -.0069860345 -.0360740205 - *0144160280 -.0325649654 3070 -.0064816337 -.0366382037 -.0085329833 -.0358419433 -.0132154408 -.0336221710 -.0201178817 -.0287411592 3080 -.0127248771 - *0347245976 -.0146627728 -.0333607746 - 0190069355 -.0301785815 -.0251823117 -.0240886717 3090 -.0185423235 -.0317724286 -.0203097274 -.0298894809 -.0241892704 -.0258527534 -.0294623417 -.0187508499 3100 -.0237606562 - *0278776909 -.0253064053 -.0255390615 -.0286105869 -.0207799025 -.0328352991 - *0128904790 C 3110 -.0282257965 -.0231642493 -.0295060669 -.0204462861 -.0321428519 -.0151167530 -.0352063206 -.0066848377 3120 -.0318074363 -.0177799191 - *0327869693 -.0147694144 - o0346855513 - *0090367152 -.0365110071 - *0003203544 Q 3130 -.0344028251 -.0118918482 -.0350558788 -.0086832956 -.0361685348 -.0027245929 -.0367171560 *0060129774 3140 -.0359396993 -.0056813447 -.0362507029 -.0023740003 -.0365539326,0036290171 - *0358255246 o0121272552 3150 -.0363782727.0006616894 -.0363421660.0039668524 -.0358370907 *0098334640 - *0338696063 o0178422295 3160 -.0357122328.0069445492 -.0353344860.0101478988 -.0340465042,0157037497 -.0309144332 *0229906058 3170 -.0339687200.0129775665 - *0332650357 *0159837907 - *0312427561.0210660312 -.0270544465.0274229312 3180 -.0312072960.0185798195 -.0302030080.0213007391 -.0275164995.0257627559 -.0224105029.0310119239 3190 -.0275179395.0235845442 -.0262471293.0259416755 -.0229855495 *0296572789 -.0171261119.0336561193 3200 -.0230181357.0278440841 -.0215224965.0297708782 -.0177911778.0326378265 -.0113630195.0352827295 3210 -.0178491525.0312342312 -.0161766388.0326779277 - 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*0105500317 3330.0344118796 -.0005920887.0343813173 -.0035499031 *0339537613 -.0088077605.0322867576 -.0160295725 3340.0337876215 -.0065372765.0334511918 -.0094064563.0323216554 -.0143965910.0296050377 -.0209987372 3350.0321435351 -.0122490479.0315150407 -.0149440909 *0297258964 -.0195197672.0260535372 -.0253125584 3360.0295354765 -.0175559636.0286373671 -.0199976307 o0262503257 -.0240262901.0217435831 - *0288464098 3370.0260481386 -.0222998465.0249107317 -.0244174723 *0220047593 -.0277845401.0168085262 -.0314995992 3380.0217923178 -.0263404837.0204527925 -.0280740025 *0171215342 - o0306861104 *0113996316 -.0331982288 0 3390.0169014099 -.0295597583.0154026051 -.0308613870 90117513985 -.0326489238.0056814572 - *0338972441 3400.0115272416 -.0318650896.0099162988 -.0327006221 *0060588713 -.0336195426 - *0001731395 -.0335816126 3410.0058353711 -.0331920815.0041622630 -.0335417591 *0002172165 -.0335746067 - *0059882134 -.0322666033 3420.0000000000 -.0335063038 -.0016840086 -.0333652416 -.0055968193 -.0325213623 -.0115900079 -.0299971632 3430 -.0058013456 -.0328041583 -.0074452580 -.0321823189 -.0112083418 - *0304972705 - *0168121358 - *0268464140 3440 -.0113932039 -.0311128081 -.0129478254 -.0300345294 -.0164495586 -.0275687118 -.0215005119 -.0229133189 3450 -.0166074723 -.0284891777 -.0180268720 -.0269922738 -.0211647861 -.0238288314 -.0255178893 -.0183195921 C 3460 -.0212884492 -.0250180603 -.0225313070 -.0231525274 -.0252150627 -.0193945938 -.0287478696 -.0132059491 3470 -.0252974718 -.0208093936 -.0263282704 -.0186357643 -.0284822342 -.0144031400 -.0310982661 -.0077278108: 3480 -.0285170118 -.0159947917 -.0293070412 -.0135821914 -.0308723905 -.0090075595 -.0325037258 -.0020505934 3490 -.0308541096 -.0107234416 -.0313822539 -.0081474091 -.0323185534 -.0033722084 -.0329275352.0036552742 3500 -.0322430445 -.0051574929 -.0324963324 -.0024976319 -.0327825400.0023322855 -.0323625596.0092194555 Q 0 0 C,,

CD TABLE 3. (Continued) CD (n 0 2 ~~~ ~ ~~~~~3j 4 5 /j67 / 3500 -.0322430445 - *0051574929 -.0324963324 - 0024976319 -.0327825400.0023322855 - 90323625596.0092194555 3510 - *0326471678.0005329188 -.0326210704.0031953827 -.0322559497.0079341770 -.0308312935.0144767807 0 3520 -.0320598454.0061748606 -.0317583169.0087595446 - *0307602502.0132657641 - 90283850243.0192721411 3530 - 90305044879.0115978470 -.0299397505.0140276223 -.0283459644.0181683977 - 0251021408.0234650570 3540 - *0280336726.0166389804 -.0272257546.0188422584 -.0250909881.0224971933 - 0210856388 *0269337852 3550 - *0247273879.0211478336 -.0237034316 *0230606598 -.0210980938 90261253036 - 0164599062.0295788475 3560 -.0206904591.0249909278 -.0194838231.0265588222 -.0164916998.0289476265 - 0113668838.0313258760 3570 -.0160492380.0280556736 -.0146984214.0292351659 -.0114140065.0308838409 - 0059617232.0321276954 3580 -.0109476596.0302536584 -.0094950840.0310134744 -.0060206180.0318806822 - 0004080722.0319655846 3590 -.0055427898.0315231850 -.0040334761.0318450529 -.0004757853.0319133940.0051268695.0308496826 3600.0000000000.0318309886.0015198178.0317100454.0050525851.0309863159 0104773730.0288185325 3610.0055120818.0311730850.0069963640.0306178743.0103980052.0291325955.0154841333.0259377807 0 3620.0108266910.0295747289.0122309787.0286067960.0154004098.0264130373.0199990041.0222980757 3630.0157839613.0270894895.0170666997.0257425888.0199109398.0229141273.0238893725.0180122312 3640.0202357237.0237974740.0213594832 o0221164188.0237963629.0187452977.0270420472 90132117436 3650.0240499252.0198027603.0249824851.0178419508.0269430000 o0140355174.0293665439.0080427702 3660 90271145358.0152301178.0278297993 o0130517980.0292600427.0089293137.0307976743.0026616914 3670.0293408290.0102211215.0298195433.0078934185.0306821637.0035823486.0312973648 -.0027696104 3680.0306659391.0049297789.0308962017.0025245861 0311713476 - 90018433161.0308556529 -.0080886104 3690.0310546230 -.0004821948 90310321615 - 0028914250.0307178891 -.0071841211.0294908381 -.0131369957 3700.0305001772 -.0058503100.0302283950 -.0081907957 0293405339 -.0122799305.0272487852 -.0177653697 3710.0290244858 -.0110122916.0285142773 -.0132141080.0270857630 -.0169788248 0242014058 -.0218376656 3720.0266772046 -.0158129760.0259465459 -.0178111313.0240262452 -.0211416211.0204443683 -.0252351395 3730.0235341091 -.0201089650.0226074398 -.0218453067.0202585103 -.0246459888.0160941075 -.0278598273 3740.0196946616 -.0237729005.0186020785 - *0251977944.0158999152 -.0273900356.0112842286 -.0296373626 3750.0152788745 - 90266972310.0140551636 -.0277709664 *0110849998 -.0292952633 0061614147 -.0305190765 3760 *0104235695 -.0287973588.0091071064 -.0294912401.0059613444 -.0303088036.0008809636 -.0304833200 3770.0052781473 -.0300140773 90039097017 -.0303111720.0006850557 - 90304048742 -.0043979100 -.0295359718 3780.0000000000 -.0303152273 -.0013785195 -.0302107519 -.0045839798 -.0295854122 -.0095168666 - o0277101233 3790 -.0052502943 -.0296965267 -.0065971167 -.0291978631 -.0096869321 -.0278798729 -.0143231826 -.0250649513 3800 -.0103138477 - e0281815533 -.0115885983 -.0273078983 -.0144708065 - *0253442022 - 0186742997 - s0216838176 3810 -.0150382623 - s0258208851 - 90162031661 -.0246025478 -.0187930230 -.0220590181 - 0224420454 -.0176716563 3820 - s0192822079 -.0226904288 -.0203032020 -.0211678008 - 0225256574 -.0181270610 - 0255164022 - e0131517291 3830 - o0229196415 -.0188889910 -.0237673643 - 0171112238 -.0255592198 -.0136699918 - 0278087136 -.0082618495 3840 -.0258435420 -.0145351702 - 90264941700 - 0125586023 - e0278058587 -.0088246375 - 0292542335 - 90031501913 3850 - 90279690500 - o0097636657 -.0284049592 -.0076500497 -.0292018966 -.0037388007 -.0298139466.0020291923 3860 -.0292359212 -.0047211193 -.0294461545 -.0025357043 -.0297096250.0014332400 - 0294756082.0071210026 3870 -.0296102220 o0004383826 -.0295907511.0026288544 - 0293183077.0065354016 - 0282539769.0119733440 3880 -.0290852206.0055580092 -.0288389960.0076873298 -.0280443669.0114144985 -.0261902358.0164422485 0 3890 -.0276814505.0104829439 -.0272182409.0124874302 -.0259307501.0159248345 - 0233506257.0203959399 3900 -.0254459490.0150650577 -.0247819765.0168854457 -.0230455013.0199325456 - 0198243325.0237187143 3910 -.0224506974.0191673567 -.0216080821.0207505432 -.0194795834.0233195666 - 0157206985.0263143200 3920 -.0187903149.0226680696 -.0177963470 90239686507 -.0153440210.0259871019 - 0111658415.0281087403 3930 -.0145790788.0254642547 -.0134653426.0264458161 -.0107664532.0278585008 -.0062987891.0290522979 3940 -.0099473658.0274748184 -.0087487432.0281109428 -.0058872029.0288814522 -.0012672439.0291210208 C 3950 -.0050376242.0286428583 - 90037912078 90289178206 -.0008549841 0290294365.0037768909 0283172343 3960.0000000000.0289372624.0012560477.0288463959.0041776228.0283023956.0086820889.0266693623 3970.0050122457.0283535202.0062398555.0279032464.0090587895.0267266032.0133017460.0242309506 3980.0098473923.0269137257.0110097362.0261212491.0136420129.0243537446.0174985543.0210789423 3990.0143598445.0246657751.0154224346.0235584563.0177905050.0212592363.0211485776.0173112626 F. 4000.0184145086.0216817883.0193462148.0202962178.0213812681.0175398408.0241449050.0130437871 0 00 0" ^ 0'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C C,,

- -I CD C TABLE 3. TABULATION OF FOURIER TRANSFORMS OF LEGENDRE POLYNOMIALS (D u1 00 01/ e2 e03/1 04 05/j 06 e 7/ 4000,0184145086,0216817883,0193462148,0202962178 0213812681.0175398408.0241449050.0130437871 4010 *0218908297.0180558060.0226647859.0164366083.0243087355.0133106503.0264007767 o0084067875 0 4020.0246863685.0139008345.0252807410.0120992428.0264878675.0087015345.0278520897.0035409640 4030.0267198120.0093453323.0271184086.0074175775.0278566149.0038531610.0284592125 -.0014068156 4040.0279333306.0045292459.0281260336.0025348115.0283776767 -.0010872900.0282080558 -.0062879418 4050.0282942121 -.0004002812.0282772236 -.0024004856.0280395042 - 0059705862.0271103725 - *0109565193 4060.0277957280 -.0052933974.0275716226 -.0072388870.0268565232 -.0106499434 0252032806 - 0152737219 4070.0264572094 -.0100020904.0260347935 -.0118346258.0248685727 -.0149854312.0225480269 - 0191119148 0 4080.0243233336 -.0143846578.0237173188 -.0160499792.0221395804 - o0188481510 90192280328 - 0223584222 4090.0214626472 -.0183099642.0206931482 -.0197593897.0187555180 -.0221240616.0153462835 -.0249188304. 4100.0179653742 -.0216613578.0170572491 -.0228531973.0148217022 -.0247173409 0110221430 -.0267197286 4110.0139405790 -.0243401308.0129226324 - 0252408757.0104595255 - 0265531822.0063876914 -.0277108077 4120.0095127722 -.0262684185.0084168478 -.0268536732.0058027179 - 0275799453.0015836987 - 0278662585 4130 o0048180667 -.0273914554.0036780559 -.0276465850.0009932549 -.0277706005 -.0032446435 -.0271854293 4140 OOOOOOOOCO -.0276791205 -.0011492006 -.0275995984 - 0038229647 - 0271234250 -.0079521037 -.0256927280 4150 -.0047948471 - 0271267304 - 0059183993 -.0267181771 - 0085005370 - 0256619351 -.0123977722 -.0234367736 4160 -.0094213032 -.0257550581 -.0104854783 -.0250329743 -.0128989366 - 0234340581 - 90164492729 -.0204888271 4170 -.0137399951 -.0236095821 -.0147131817 -.0225987869 -.0168867323 - 0205105743 -.0199866994 -.0169405514 4180 -.0176215393 - 0207589922 -.0184751776 -.0194927844 - 0203455077 - 0169828788 -.0229061584 - o0129011705 4190 -.0209504122 -.0172929986 -.0216598269 -.0158120718 -.0231733722 -.0129601320 -.0251228160 -.0084941159 4200 -.0236283812 -.0133195174 -.0241734899 - 90116706615 -.0252879560 -.0085658905 - 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(CD C TABLE 4. TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS (D -< A0 B A0B/j B 2/j A3 B3/j A3 B 4/j 0 2,0000000000.0000000000.00000000000000.0000000000.0000000000.0000000000.0000000000 10 1.9898615401 -.1108901168.0061605620.0553166142 -.0015365726 - *0368619290 *0006826283.0276422981 20 1.9596310721 -.2204584956.0244953884.1092054467 -.0060669693 - *0726786249 *0026918009 *0544762306 O 30 1.9098593171 - 3274044544.0545674091.1602679147 -.0133556596 -.1064317886 *0059128771 *0797158671 40 1.8414508579 -.4304690317.0956597848.2071632215 -.0230181357 -.1371563398.0101597289.1026195525 50 1.7556445404 -.5284548784.1467930218.2486357335 -.0345327408 -.1639654189.0151819832.1225105708 60 1.6539866863 -.6202450073.2067483358.2835405748 -.0472567625 -.1860735022 *0206748336.1387960856 70 1.5382977489 -.7048200595.2740966898 *3108669067 -.0604463430 -.2028170733 *0262911021.1509838492 80 1.4106331970 -.7812737706.3472327869.3297584097 -.0732796466 -.2136723496 *0316551629.1586962347 90 1.2732395447 -.8488263632.4244131816.3395305453 -.0848826363 -.2182696362.0363782727.1616812120 100 1.1285065576 -.9068356266.5037975704.3396842481 -.0943567356 -.2164039564 *0400748067.1598199688 -. 110.9789167493 -.9548054993.5834922496.3299157725 -.1008075972 -.2080416991.0423788646.1531309630 c 120.8269933431 -.9923920118.6615946745.3101225037 -.1033741679 -.1933231192 *0429606931.1417702874 Q 130.6752479002 -1.0194065073.7362380331.2804046293 - 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-.a CD c TABLE 4. TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS CD A0 B1/j A1 B2/j A3 B3/j A3 "4/j 500.1473160686,0609303593 -.1692509980 -.2202399697 ~3058888468 -.9561379455.ob53129125.1975923424 510.1123446657.0452931063 -.1283304679 -.1580573228.2239145406 -.9822133631.9276459540.1597094899 0 520.0753704259.0296414784 -.0856309376 -.1002084071.1447454769 -.9983026216.9613284505.1137850300 530.0375445574.0144134840 -.0424397029 -.0473463420.0697043368 -1.0042291885.9856323516.0603242235 540.0000000000.0000000000.0000000000.0000000000.0000000000 -1.0000000000 1.0000000000.0000000000 550 -.0361793007 -.0132608322.0405192094.0414314722 - *0632980825 -.9858029648 1.0040585753 -.0663594400 560 -.0699868240 -.0250877378.0780507397.0766812159 -.1192818913 -.9620007081.9976303640 -.1377865598 -. 570 -.1005189114 -.0352589412.1116533139.1056143401 - *1672227052 -.9291207489.9807365682 -.2131935982 580 -.1269966109 -.0436133098.1405317760.1282248180 -.2065844289 -.8878423779.9536084800 -.2913944215 590 -.1487834356 -.0500500502.1640529425.1446297562 -.2370321004 -.8389805768.9166639636 -.3711286169 -. 600 -.1653986686 -.0545270336.1817567787.1550612518 -.2584354197 -.7834673777.8705193086 -.4510872781 610 -.1765259712 -.0570578500.1933627138.1598560960 -.2708672737 -.7223311169.8159666320 -.5299398943 620 -.1820171867 -.0577077217.1987710414.1594436267 -.2745973570 -.6566740745.7539591225 -.6063617324 Q 630 -.1818913635 -.0565884242.1980594847.1543320660 -.2700811155 - *5876490206.6855905241 -.6790610905 640 -.1763291496 -.0538523808.1914751317.1450937003 -.2579443560 -.5164352043.6120713533 -.7468058102 650 -.1656628345 -.0496861103.1794220650.1323492771 -.2389639726 -.4442143232.5347024261 - *8084484520 660 -.1503624260 -.0443032148.1624451210.1167520014 -.2140453359 -.3721470044.4548463387 -.8629495754 670 -.1310182493 -.0379370969.1412103052.0989715056 -.1841969687 -.3013503025.3738975976 -.9093986109 680 -.1083206387 -.0308335958.1164824729.0796781621 -.1505031951 -.23z8766893.2932521273 -.9470318698 690 -.0830373616 -.0232437239.0891009418.0595280826 -.1140954917 -.1676949644.2142768990 -.9752473109 700 -.0559894592 -.0154166780.0599537478.0391491224 -.0761232936 -.1066734656.1382804184 -.9936157549 710 -.0280262189 -.0075932813.0299512761.0191281750 -.0377250118 -.0505658933.0664847856 -1.0018883279 720.0000000000.0000000000.00C0000000.0000000000.0000000000.0000000000.0000000000 -1.0000000000 730.0272583773.0071563452 -.0290229555 -.0177622156.0360178260.0445306768 -.0601988779 -.9880691785 740.0529630019.0136930743 -.0562937497 -.0337544108.0693840667.0826719359 -.1132911714 -.9663934054 750.0763943727.0194552732 -.0810636382 -.0476472179.0992650373.1142108562 -.1586261891 -.9354412982 760.0969184662 *0243185000 -.1026781109 -.0591894919.1249555939.1390746102 -.1957346366 -.8958409579 770.1140028922.0281905457 -.1205928899 -.0682098030.1458931897.1573262615 -.2243355951 -.8483651470 780.1272297451.0310122504 -.1343864183 -.0746159407.1616678716.1691577293 -.2443389423 -.7939136094 790.1363048638.0327573967 -.1437685745 -.0783925243.1720280395.1748801651 -.2558432045 -.7334929664 800.1410633197.0334317282 -.1485854585 -.0795968575.1768819056.1749120382 -.2591289455 -.6681946723 810.1414710605.0330711570 -.1488202065 -.0783532028.1762947062.1697652726 -.2546479089 -.5991715505 820.1376227509.0317392469 -.1445899027 -.0748456818.1704818307.1600298165 -.2430082398 -.5276134555 830.1297359547.0295240695 -.1361387650 -.0693100362.1597981391.1463570473 -.2249562024 -.4547226190 840.1181419062.0265345458 -.1238278803 -.0620245007.1447238350.1294424363 -.2013549009 -.3816892353 850.1032732083.0228963931 -.1081218562 -.0533000535.1258473486.1100078965 -.1731605778 -.3096678268 860.0856488771.0187478029 -.0895728358 -.0434703153.1038457533.0887842378 -.1413971194 -.2397549037 870.0658572178.0142349763 -.0688023853 -.0328813656.0794633001.0664941348 -.1071294395 -.1729683910 880.0445370698.0095076440 -.0464818152 -.0218817365.0534886893.0438359892 -.0714364268 -.1102292478 890.0223579948.0047146910 -.0233115278 -.0108128325.0267317248.0214690356 -.0353841513 -.0523456452 0 900.000000COOC.000000000C.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000 910 -.0218666103 -.0045013834.0227569939.0102555498 -.0259237509 -.0200284293.0337516123.0462609076 920 -.0426006755 -.0086666730.0442963289.0196837943 -.0503030298 -.0381471081.0649913694.0860325837 930 -.0616083651 -.0123882767.0640060960.0280517892 -.0724671222 -.0539677672.0929444880.1190561652 940 -.0783596110 -.0155761821.0813422841.0351682922 -.0918283186 -.0671880718.1169570139.1452184685 950 -.0924023442 -.0181597527.0958431395.0408870510 -.1078963846 -.0775943732.1365086195.1645488244 C 960 -.1033741679 -.0200889071.1071408380.0451087278 -.1202899408 -.0850621724.1512216399.1772128592 970 -.1110111778 -.0213346743.1149701895.0477814659 -.1287445053 -.0895543738.1608661900.1835034474 980 -.1151537304 -.0218891358.1191741839.0489001397-.1331170470 -.0911174553.1653613078.1838291225 990 -.1157490495 -.0217647785.1197062820.0485043636 -.1333869999 -.0898757326.1647721764.1787002870 1000 -,1128506558 -.0209933010.1166294499,0466753631 -.1296537865 -.0860239330.1593035796.1687136050 00~ 0 0" 0) (/)

C TABLE 4. (Continued) w A0 B1/j A B2/j A3 B/ A3 -4 /. 1000 - 1128506558 -.0209933010 1166294499.0466753631 - *1296537865 -.0860239330.1593035796.1687136050 1010 -.1066146955 -.0196239259.1101120288.0435318397 -.1221309948 -.0798183321.1492898433.1545349941 1020 -.0972933345 -.0177212859.1004206202.0392249807 -.1111374453 -.0715667316.1351816042.1368816568 0 1030 - *0852254631 -.0153629566.0879102517.0339327840 - *0970854653 - 0616175778.1175298243.1165036008 1040 -.0708250330 -.0126367196.0730121575.0278538785 -.0804667602 -.0503485298.0969675388.0941651007 1050 - 0545674091 -.0096376416 90562195762.0212010285 -.0618363331 -.0381547921.0741898736.0706265387 1060 -.0369741712 -.0064650585.0380720113.0141945108 -.0417949484 -.0254375187.0499329071.0466270447 1070 - 0185968368 -.0032195513.0191384436 0070555516 -.0209706673 -.0125925845.0249519729 0228683210 1080 0000000100o.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000 1090.0182556105.0030992046 -.0187674057 -.0067675971.0204907801.0119857806 - 0z41935201 -.0213931657 1100.0356296559.0059907200 -.0366099555 -.0130593703.0399036316.0230444420 -.0469423820 -.0408020350 -. 1110.0516178194.0085965161 -.0530118490 -.0187088260.0576855467.0328970983 -.0676218131 -.0578015430 1120.0657661021.0108497923 -.0675098187 -.0235747621.0733437042.0413121505 -.0856844604 -.0720567727 0 1130.0776833867.0126965053 -.0797058389 -.0275443230.0864585693 0481096139 -.1006738217 -.0833265701 1140.0870519309.0140964774 -.0892776905 -.0305351388.0966946063.0531638578 - 1122348109 -.0914647015 1150.0936355151.0150240692 -.0959871086 -.0324965238.1038083400.0564047482 -.1201212230 -.0964186128 1160.0972850481.0154684115 -.0996853188 -.0334097336.1076535861.0578172240 -.1241999627 -.0982259093 1170.0979415034.0154332066 -.1003158429 -.0332873084.1081837522.0574393780 -.1244519856 -.0970087272 1180.0956361489.0149361162 -.0979145396 -.0321715522.1054511988.0553591516 -.1209699980 -.0929662153 1190.0904881029.0140077702 -.0926069253 -.0301322217.0996037330.0517097865 -.1139530480 -.0863653855 1200.0826993343.0126904349 -.0846028996 -.02T2635168.0908783894.0466642012 -.1036982248 -.0775306259 1210.0725472951.0110363909 -.0741890722 -.0236804809.0795927274.0404284884 -.0905897609 -.0668321915 1220.0603754380.0091060757 -.0617189573 - 0195149340.0661339431.0332347414 -.0750858973 -.0546740049 1230.0465819346.0069660522 -.0476013568 -.0149110682.0509461496.0253334324 -.0577039293 -.0414811073 1240.0316069528.0046868668 -.0322873044 - 0100208407 0345162291 0169855663 -.0390038929 -.0276870952 1250.0159188923.0023408606 -.016255976? 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00 -- C TABLE 4. TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS (D (n aA0 13 A1' /j A3 B3/j A3 B /j 1500 ~066159467,90080551300 -,0671260830 -.0168487608 *0702031700 *0273637503 -.0760104176 -.0412637011 1510.0713118162.0086232823 - *0723307571 -.0180260885.0756094267.0292419688 - *0817692090 - 90440087453 0 1520.0742438525.0089170843 -.0752998230 -.0186290518.0786559963.0301859293 -.0849678010 -.0453417813 153C.0748964438.0089350143 -.0759476220 -.0186555245.0792859790.0301954042 -.0855536453 -.0452707394 1540.0732796466.0086837885 -.0742946349 -.0181205335.0775156155.0292977664 -.0835528894 -.0438444162 155C.C694715112.0081779468 -.0704212083 -.0170553492.0734327536.0275463806 -.0790683148 -.0411497301 1560 *0636148725.0074392205 -.0644732441 -.0155061252 *0671932091.0250182860 -.0722750485 - *0373079704 * 1570.0559122465.0064957044 -.0566569769 -.0135321379.0590151571.0218112560 -.0634142072 -.0324701790 1580.0466190091.0053808629 -.0472320188 -.0112036895.0491717484.0180403386 -.0527846944 -.0268118224 159C.0360350815.0041324039 -.0365029008 -.0085997418.0379821928.0138339910 -.0407334180 -.0205269250 1600.0244953884.0027910555 -.0248093821 -.0058053587.0258015941.0093299310 -.0276442401 -.0138218493 1610.0123593884.0013992854 -.0125158303 -.0029090362.0130098562.0046708334 -.0139260034 -.0069089083 1620.000000000C.0000000000.0000000000.0030000000.0000000000.0000000000.0000000000.0000000000 163C -.0122077395 -.0013647364.0123584466.0028344484 -.0128337527 -.0045428708.0137127397.0066995572 1640 -.0238979399 -.0026549269.0241893343.0055114624 -.0251077731 -.0088256946.0268039613.0129968162 1650 - 0C347247149 -.0038337758.0351429449.0079549850 -.0364603479 -.0127276815.0388901379.0187164653 1660 -.0443723098 -.0048687009.0449002412.0100978265 -.0465621999 -.0161425887.0496235134.0237054285 1670 -.C525642078 -.0057321969.0531820494.0118834286 -.0551259050 -.0189814610.0587019257.0278366895 1680 -.0590709531 -.0064025291.0597569383.0132672729 -.0619139401 -.0211747940.0658771369.0310122504 1690 -.0637164748 -.0068642390.0644475772.0142178996 -.0667451398 -.0226740709.0709614442.0331651651 1700 -.0563827387 -.0071084540.0671353985.0147175178 -.0694993895 -.0234526455.0738324026.0342606140 1710 -.0670126076 -.0071329946.0677634492.0147621976 -.0701204387 -.0235059608.0744355426.0342960147 1720 -.0656108464 -.0069422823.0663373643.0143616502 -.0686167732 -.0228511152.0727850337.0333001935 1730 -.0622432615 -.0065470534.0629244578.0135386104 -.0650605446 -.0215258042.0689622987.0313316656 1740 -.0570340237 -.0059638943.0576509782.0123278511 -.0595846135 -.0195866844.0631126499.0284760979 1750 -.0501612726 -.0052146137.0506976328.0107748651 -.0523778163 -.0171072224.0554400726.0248430500 1760 -.0418511559 -.0043254751.0422935340.0089342629 -.0436786187 -.0141751028.0462003351.0205621064 1770 -.0323704969 -.0033263152.0327087662.0068679388 -.0337673658 -.0108892834.0356926512.0157785279 1780 -.0220183267 -.0022495765.0222458119.0046430643 -.0229573734 -.0073567907.0242501620.0106485613 1790 -.0111165449 -.0011292842.0112301042.0023299729 -.0115851428 -.0036893570.0122295353.0053345508 1800.0C0000000.000000000 0.0000000000C 0000000000.0000000000.0000000000.0000000000.0000000000 1810.0109937102.0011042176 -.0111035219 -.0022766571.0114465259.0036003526 -.0120678485 -.0051952725 1820.0215344074.0021508146 -.0217471253 -.0044329973.0224112640.0070060963 -.0236131392 -.0100997450 1830.0313091691.0031096760 -.0316150389 -.0064071322.0325695886.0101199493 -.0342953837 -.0145744431 1840.0400315404.0039539682 -.0404183416 -.0081440087.0416249334.0128556362 -.0438043902 -.0184967162 1850.0474498524.0046608657 -.0479033421 -.0095968908.0493173555.0151401842 -.0518691494 -.0217634391 1860.0533544092.0052121429 -.0538588102 -.0107285623.0554309052.0169157729 -.0582654401 -.0242935621 1870.0575833382.0055046173 -.0581218575 -.0115122214.0597995947.0181410960 -.0628219434 -.0260299502 1880.0600269446.0058004335 -.0605823052 -.0119320477.0623118048.0187922025 -.0654247048 -.0269404748 0 1890.0606304545.0058271833 -.0611854243 -.0119834310.0629130128.0188628081 -.0660198282 -.0270183458 1900.0593950820.0056778617 -.0599329847 -.0116728641.0616067825.0183640773 -.0646143461 -.0262816898 1910,0563774044.0053606637 -.0568825978 -.0110175084.0584540028.0173238954 -.0612752597 -.0247724084 1920.0516870839.0048886306 -.0521453931 -.0100444536.0535704190.0157856630 -.0561268017 -.0225543639 1930.0454830192.0042791607 -.0458821119 -.0087896981.0471225484.0138066591 -.0493460222 -.0197109632 1940.0379680589.0035534000 -.0382977558 -.0072968884.0393221207.0114560307 -.0411568510 -.0163422238 1950.0293824510.0027355349 -.0296349620 -.0056158569.0304192246.0088124757 -.0318228289 -.0125614206 C 1960.0199962354.0018520088 -.0201663179 -.0038010085.0206943799.0059616945 -.0216387431 -.0084914201 3 1970.0101008200.0009306875 -.0101858574 -.0019096051.0104497834.0029936902 -.0109214254 -.0042608188 1980.OCO000o 000.00000000 000000000.0OOOOOOCO0.0000000000.0000000000.0000000000.0000000000 1990 -.0099993042 -.0009119207.0100817895.0018701218 -.0103376177 -.0029290588.0107941242.0041627693 2000 - e0195963107 -.0017780703.0197563371.0036454485 - e0202524914 -.0057070477.0211372136.0081050918 00 0 (D C,,

CD C TABLE 4. (Continued) (D 0A B/j A1 B/ A3 B3/J A3 B4/j - 2000 -,0195963107 -,0017780703.0197563371,0036454485 -,0202524914 -.0057070477.0211372136.0081050918 2010 -.0285053629 -.0025733564.0287358128.0052746402 -.0294500746 -.0082538938.0307228269.0117139340 0 2020 -.0364643734 -.0032753079.0367562325.0067117780 -.0376605319 -.0104981362.0392708059.0148887661 2030 -.0432424764 -.0038646939.0435851586.0079176213 -.0446465865 -.0123788689.0465353774.0175442619 2040 -.0486466672 -.0043260332.0490283761.0088606430 -.0502103101 -.0138473304.0523121371.0196124904 2050 -.0525272402 -.0046479825.0529353557.0095178137 -.0541986611 -.0148681007.0564437155 *0210445445 2060 -.0C547818717 -.0048235938.0552033508.0098751182 -.0565076205 -.0154198771.0588239754.0218115731 -" 2070 -.0C53582411 -.0048504364.0557800182.0099277937 -.0570848139 -.0154958160.0594006278.0219051965 t 2080 -.0542551230 -.0047305818.0546645002.0096802888 -.0559305576 -.0151034386.0581762080.0213373089 2090 -.0515219342 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2190.0261624564.0021649643 -.0263403987 -.0044201180.0268890511.0068686249 -.0278560899 -.0096437247 2200.0334809247.0027578098 -.0337065637 -.0056294354.0344021050.0087449078 -.0356274021 -.0122717939 2210.0397204647.0032567555 -.0399857208 -.0066466719.0408031805.0103216990 -.0422425088 -.0144772209 2220.3447023429.0036485000 -.0449981672 -.0074448013.0459096079.0115573604 -.0475135927 -.0162023186 2230.0482873733.0039231964 -.0486040438 -.0080038617.0495794765.0124212622 -.0512952126 -.0174049178 2240.0503797570.0040746047 -.0507071871 -.0083114231.0517155215.0128944888 -.0534882499 -.0180593155 2250.0509295818.0041006105 -.0512576306 -.0083628213.0522676332.0129701821 -.0D40424255 -.0181567137 2260.0499339185.004002427] -.0502526959 -.0081611563.0512339257.0126535204 -.0529573260 -.0177051495 2270.0474364045.0037852P50 -.0477366492 -.0077170592.0486603453.0119613385 -.0502819230 -.0167289258 2280.0435259654.0034578119 -.0437989506 -.00/0482387.0446388452.0109214086 -.0461126149 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K(D TABLE 4. TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS (D A0 1 i A1 B2 / A3 B3 / A3 34/j. 2500 -.0156770486 -.0011346294,0157587419.0023052976 -,0160090114 -.0035519629.0164442726.0049233525 < 2510 -.0079277352 -.0005714617.0079687165.0011609272 -.0080942427 -.0011883419.0083124781.00z4779914 2520,0000000000.0000000000.0000000000.0000000000.0000000000.000000000 00 000 000000.0000000000 2530.0078650654.0005624167 -.0079050792 -.0011422674.0080276014.0017588353 -.0082404690 -.0024355296 2540.0154301659.0010989955 - *0155080475 -.0022317848.0157464819.0034357168 -.0161605939 -.0047560654 2550.0224689331.0015939847 -.0225814497 -.0032365925.0229258639.0049815197 -.0235238432 -.0068937620 2560.0287726697.0020331298 -.0289156241 -.0041277852.0293531395.0063518586 -.0301125148 -.0087874113 2570.0341565386.0024040777 -.0343248876 -.0048803301.0348401346.0075083399 -.0357341364 -.0103841622 2580.0384648067.0026967174 -.0386529497 -.0054737563.0392285866.0084196155 -.040Z270518 -.0116409620 ^ 2590.0415756148.0029034486 -.0417773989 -.0058926964.0423946769.0030622076 -.0434650326 -.0125z56808 2600.0434040984.0030193707 -.0436131317 -.0061272677.0442524893.0094i1086i -.0453607855 -.0130170916 2610.0439048119.0030423884 -.0441146318 -.0061732826.0447562991.0094899842 -.0458682569 -.0131092875 2620.0430727694.0029732316 -.0432770372 -.0060322855.0439016330.009z714451 -.0449836779 -.0128037302 2630.0409432861.0028153899 -.0411359744 -.0057114174.0417250771,0087766071 -.0427453271 -.01Z1169347 0 2640.0375906065.0025749663 -.0377661724 -.0052231159.0383028496.0080247332 -.0392320292 -.0110758037 2650.0331253687.0022604542 -.0332789090 -.0045846598.0337481904.0070425058 -.0345604453 -.0097174403 2660.0276909903.0018824464 -.0278183739 -.0038175774.0282076549.0058631100 -.0288812458 -.0080878705 2670.0214590935.0014532855 -.0215570678 -.0029469348.0218564330.0045251373 -.0223742897 -.0062405206 2680.0146241125.0009866674 -.0146903812 -.0020005308.0148928405.0030713438 -.0152429653 -.0042344969 2690.0073972548.0004972099 -.0074305253 -.0010080210.0075321572 *0015473040 -.0077078660 -.0021327245 2700.0000000000.0000000000.000)000000.0000000000.0000000000.0000000000.0000000000.0000000000 2710 -.0073426625 -.0004898657.0073751997.0009929311 -.0074745648 -.0015236090.0076462599.0020989793 2720 -.0144090520 -.0009577344.0144724315.0019410829 -.0146659600 -.00o9779951.0150002716.0041015534 2730 -.0209874650 -.0013898309.0210791022.0028165555 -.0213588789 -.0043204056.0218420508.0059489453 2740 -.0268824943 -.0017736577.0269990119.0035940488 -.0273547051 -.0055121035.0279688212.0075679621 2750 -.0319208098 -.0020983521.0320581565.0042515844 -.0324773809 -.0065194684.0332009963.0089725023 2760 -.0359562323 -.0023549882.0361098185.0047711154 -.0365785517 -.0073149288.0373874140.010064d277 2770 -.0388739503 -.0025368172.0390387977.0051390112 -.0395418363 -.0078776984.0404096749.0108365653 2780 -.0405937611 -.0026394384.0407646600.0053464015 -.0412861008 -.0081942978.0421854592.0112694136 2790 -.0410722434 -.0026608977.0412439142.0053893737 -.0417676464 -.0082588511.0426707307.0113555366 2800 -.0403038056 -.0026017109.0404710585.0052690180 -.0409812508 -.0080731485.0418607700.0110976388 2810 -.0383205845 -.0024648132.0384784729.0049913221 -.0389600421 -.0076464797.0397900148.0105087231 2820 -.0351912061 -.0022554364.0353351701.0045669214 -.0357742173 -.0069952444.0365307210.0096115449 2830 -.0310184548 -.0019809191.0311444496.0040107118 -.0315286513 -.0061423558.0321904944.0084377817 2840 -.0259359276 -.0016504564.0260405340.0033413423 -.0263594780 -.0051164572.0269087751.0070269484 2850 -.0201037823 -.0012747976.0201842958.0025806004 -.0204297529 -.0039509789.0208523888.0054250945 2860 -.0137037138 -.0008659014.0137582111.0017527135 -.0139243351 -.0026830649.0142103065.0036833247 2870 -.0069333155 -.0004365593.0069606955.0008835866 -.0070441486 -.0013524036 *0071877747.0018561901 2880.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000 2890.0068853340.0004305144 -.0069121481 -.0008712072.0069938581.0013330752 -.0071344208 - *0018288935 2900.0135146970.0008420875 -.0135669645 -.0017039447.0137262212 *0026069195 -.0140001234 -.0035757879 0 2910.0196892713.0012225708 -.0197648942 -.0024736440.0199952891 *0037839810 -.0203914534 -.0051892476 2920.0252253542.0015609190 -.0253215753 -.0031579759.0256146937.0048301523 -.0261186016 -.0066226005 2930.0299598044.0018475067 -.0300733031 -.0037374903.0304190183.0057157468 -.0310132185 -.0078352698 2940.0337548303.0020743980 -.0338818343 -.0041961607.0342686459.0064163277 -.0349333395 -.0087939035 2950.0365019805.0022355626 -.0366383877 -.0045218191.0370537957.0069133690 -.0377674785 - *0094732721 2960.0381252215.0023270309 -.0382667302 -.0047064685.0386976299.0071947289 -.0394377734 -.0098569090 C 2970.0385830165.0023469853 -.0387252580 -.0047464661.0391583457.0072549262 -.0399020940 -.0099374794 2980.0378693476.0022957864 -.0380080192 -.0046425739.0384301954.0070952117 -.0391550571 -.0097168706 2990.0360136597.0021759322 -.0361446522 -.0043998755.0365434106.0067234382 -.0372279261 -.0092060043 3000.0330797337.0019919551 -.0331992510 -.0040275650,0335630415.0061537330 -.0341874057 -.0084243804 0 0 (D Lf)

C D TABLE 4. (Continued) (i1 ~~~~~~~~~~~~~~~~ -A0 B1/j A1 B/J A3 B3/j A3 B4/J, 3000,0330797337.0019919551 -.0331992510 -.0040275650.0335630415.0061537330 - ~0341874057 -.0084243804 0 3010.0291635306.0017502576 -.0292681971 -.0035386151.0295867536.0054059876 -.0301333755 -.0073993706 3020.0243900776.0014588959 -.0244770317 -.0029493362.0247416541.0045051806 -.0251956397 -.0061652854 3030.0189094982.0011273149 -.0189764674 -.0022788418.0191802517.0034805576 -.0195297956 -.0047622466 3040.0128923097.0007660461 -.0129376677 -.0015484354.0130756770.0023646945 -.0133123541 -.0032349009 3050.0065241362.0003863767 -.0065469388 -.0007809419.0066163129.0011924727 -.0067352625 -.0016310155 ~. 3060.0000000000.0000000000.0000000000,0000000000.0000000000.0000000000.0000000000.0000000000 I 3070 -.0064816337 -.0003813416.0065039925.0007706585 -.0065720041 -.0011764918.0066885736.0016086008 j 3080 -.0127248771 -.0007462103.0127684868.0015079240 -.0129011280 -.0023017375.0131284285.0031465976. 3090 -.0185423235 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3290.0266815280.0014641624 -.0267616341 -.0029549396.0270048648.0045005842 -.0274202257 -.0061326394 3300.0300724852.0016452122 -.0301622241 -.0033201472.0304346831.0050563452 -.0308998871 -.0068892191 3310.0325319766.0017743577 -.0326284673 -.0035805757.0329214047.0054524502 -.0334215002 -.0074279014 3320.0339911614.0018483274 -.0340913719 -.0037296412.0343955803.0056789199 -.0349148411 -.0077353839 3330.0344118796.0018655524 -.0345127203 -.0037641967.034C188198.0057310107 -.0353412327 -.0078052999 3340.0337876215.0018261937 -.0338860391 -.0036845852.0341847624.0056092925 -.0346945131 -.0076385207 3350.0321435351.0017321160 -.0322366040 -.0034945869.0325190728.0053195670 -.0330010175 -.0072430383. 3360.0295354765.0015868117 -.0296204843 -.0032012646.0298784692.0048726285 -.0303185776 -.0066336364 3370.0260481386.0013952758 -.0261226637 -.0028147107.0263488198.0042838787 -.0267345761 -.0058313607 3380.0217923178.0011638383 -.0218542974 -.0023477080.0220423696.0035728063 -.0223631212 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h)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0) C TABLE 4. TABULATION OF FOURIER TRANSFORMS OF TRIGONOMETRIC FUNCTIONS (D () A, 1 B/A ABB/j A^ ^^^ le A 0 BIj A1 B2/j A3 3/j A3 B4/j 3500 - 0322430445 -.0016626112.0323285502.0033518891 - *0325878102 -.0050959455 0330292761.0069259498 3510 - 0326471678 -.0016786283 90327332513.0033840254 -.0329942473 -.0051444022.0334386143.0069910201 0 3520 - 0320598454 -.0016437221.0321438994 *0033135061 -.0323987267 -.0050368098 *0328325378 *0068440425 - 3530 - 0305044879 -.0015595246.0305840104.0031436348 -.0308250853 -.0047782251.0312354346.0064919578 3540 -.0280336726 -.0014291360.0281063406.0028806735 -.0283266225 -.0043782001.0267015337.0059478095 3550 - *0247273879 - 0012570147.0247911239.0025336210 -.0249843185 -.0038504417 *0253130889.0052302781 3560 - 0206904591 -.0010488281.0207434897 *0021139104 - 0209042253 - 0032123517.0211777259 *0043630532 3570 -.0160492380 -.0008112677.0160901422.0016350369 -.0162141156 -.0024844596.0164250386.0033740614 3580 -.0109476596 -.0005518360.0109754056.0011121279 -.0110594937 -.0016897687 0112025409.0022945753 3590 -.0055427898 -.0002786119.0055567592.0005614689 -.0055990931 -.0008530348.0056711014.0011582337 3600.0000000000.0000000000.0000000000 0000000000.0000000000.0000000000.0000000000.0000000000 3610.0055120818.0002755256 - 0055258199 -.0005552026.0055674483.0008433933 - 0056382404 -.0011449054 3620.0108266910.0005396781 -.0108535258 - 0010874421.0109348344 0016517869 -.0110730902 - 0022420696 3630.0157839613 0007846050 -.0158228673 -.0015809003 0159407448 0024011642 - 0161411593 -.0032589139 3640.0202357237.0010031206 -.0202853286 -.0020211046.0204356132.0030695585 -.0206910978 -.0041656548 3650.0240499252.0011889151 -.0241085566 -.0023953491.0242861788 0036376921 -.0245881043 -.0049361692 3660.0271145358.0013367349 -.0271802769 - 90026930584.0273794268.0040895275 -.0277179088 -.0055487392 3670 90293408290.0014425298 -.0294115798 -.0029060823.0296258943.0044127139 -.0299901112 -.0059866574 3680.0306659391.0015035617 -.0307394828 -.0030289154.0309622460.0045989189 -.0313407804 -.0062386726 3690.0310546230.0015184729 -.0311286949 -.0030588337.0313530458.0046440348 -.0317342379 -.0062992680 3700.0305001772.0014873124 -.0305725330 -.0029959467.0307916748.0045482561 -.0311639768 -.0061687625 3710.0290244858.0014115187 -.0290929692 -.0028431628.0293003722.0043160256 -.0296526944 -.0058532359 3720.0266772046.0012938618 -.0267398108 - 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o0020417811.0148596288.0027637397 3940 - 0099473658 - 0004553987.0099681709 0009165482 - 0100311114 -.0013894443.0101377973.0018805939 3950 -.0050376242 -.0002300403.0050481070.0004629708 -.0050798190 -.0007018041.0051335670.0009498083 C 3960.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000.0000000000 3970.0050122457.0002277236 -.0050225707 -.0004582793.0050538025.0006946179 -.0051067277 -.0009399379 3980.0098473923.0004462723 -.0098675755 -.0008980665.0099286245.0013611350 -.0100320689 -.0018417104 3990.0143598445.0006491336 -.0143891287 -.0013062589 o0144777024.0019796987 - o0146277736 -.0026784680 4000.0184145086.0008303343 - o0184518736 -.0016708396.0185648841.0025321063 -.0187563429 -.0034256010 00 Q 0" -9. 0 (D()

(D C c TABLE 4. (Continued) (D (n A0 B1/j AI B2 /j A3 B3/j A3 B4 /j 4000.0184145086.0008303343 -.0184518736 - 0016708396.0185648841.0025321063 -.0187563429 -.0034256010 4010 *0218908297.0009846147 -.0219350268 -.0019812295.0220686957.0030023373 -.0222951345 -.0040614588 4020 *0246863685.0011075804 -.0247359616 -.0022285920.0248859439.0033770146 -.0251399973 -.0045679724 4030.0267198120.0011958264 -.0267732236 -.0024060816.0269347473.0036457810 -.0272083286 -.0049311628 4040.0279333306.0012470298 -.0279888914 -.0025090314.0281569077.0038015817 -.0284414628 -.0051415202 4050.0282942121.0012600094 -.0283502125 -.0025350713.0285195518.0038408433 -.0288063245 -.0051942460 4060.0277957280.0012347499 -.0278504706 -.0024841773.0280159999.0037635471 -.0282962983 -.0050893488 4070.0264572094.0011723908 -.0265090596 -.0023586490.0266658369.0035731938 -.0269312941 -.0048315961 4080.0243233336.0010751809 -.0243707681 -.0021630166.0245141880.0032766630 -.0247570096 -.0044303215 4090.0214626472.0009463994 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The University of Michigan Willow Run Laboratories REFERENCES 1. R. C. Spencer, P. Austin, E. Chisholm, E. Fine, and J. Schwartz, Tables of Transforms of Fourier Series, Power Series, and Polynomials, Massachusetts Institute of Technology, Radiation Laboratory, Report Number S-58, 1945. 2. National Bureau of Standards, Tables of Spherical Bessel Functions, Mathematical Tables Project, Columbia University Press, 1947. 3. Elmer G. Gilbert and Joseph Otterman, "The Synthesis of Linear Filters with Real or Imaginary Transfer Functions, " Trans. AIEE, Part I: Communications and Electronics, July 1960, Vol. 79. 24

f+ + UNCLASSIFIED UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functions- A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functionsRIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables RIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's series- Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's seriesincl. tables, 3 refs. Tables incl. tables, 3 refs. Tables (Rept. no. 36943-15-T) Unclassified report 4. Trigonometry- (Rept. no. 36943-15-T) Unclassified report 4. TrigonometryThis report presents tables of Fourier transforms of functions which I ble This report presents tables of Fourier transforms of functions which I. Gilbert, Elmer G., Otterman, I. Gilbert, Elmer 0., Otterman, are trigonometric functions or Legendre polynomials on the interval Joseph, lmand Ror d an, are trigonometric functions or Legendre polynomials on the interval J, an rdan are trigonometric functions or Legendre polynomials on the interval Joseph, and Riordan, (-1, 1) Joseph, and Riordan, (-1, 1) and are zero elsewhere. John F. (-1, 1) and are zero elsewhere. John F. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~John F. (over) (over) UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functions- A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functionsRIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables RIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's series- Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's seriesincl. tables, 3 refs. Tables incl. tables, 3 refs. Tables (Rept. no. 36943-15-T) Unclassified report 4. Trigonometry- (Rept. no. 36943-15-T) Unclassified report 4. TrigonometryTables Tables This report presents tables of Fourier transforms of functions which Tabs This report presents tables of Fourier transforms of functions which le are trigonometric functions or Legendre polynomials on the interval. Glbert Elmer., Oterman, are trigonometric functions or Legendre polynomials on the interval (-1, 1) and are zero elsewhere. Jo h, and Riordan, (-1, 1) and are zero elsewhere. John F. John F. (over) (over) UNCLASSIFIED UNCLASSIFIED + + +

UNCLASSIFIED UNCLASSIFIED UNITERMS UNITERMS Fourier transforms Fourier transforms Trigonometric functions Trigonometric functions Legendre polynomials Legendre polynomials UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNITERMS UNITERMS Fourier transforms Trigonometric functions Trigonometri fun Legendre polynomials Legendre polynomials UNCLASSIFIED UNCLASSIFIED

UNCLASSIFIED UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functions- A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functionsRIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables RIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables Giltert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's series- Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's seriesincl. tables, 3 refs. Tables incl. tables, 3 refs. Tables (Rept. no. 36943-15-T) Unclassified report 4. Trigonometry- (Rept. no. 36943-15-T) Unclassified report 4. TrigonometryThis report presents tables of Fourier transforms of functions which I. Tables This report presents tables of Fourier transforms of functions whichr are trigonometric functions or Legendre polynomials on the interval I. GierE e or an are trigonometric functions or Legendre polynomials on the interval and Riord an, (-1, 1) and are zero elsewhere. Joseph, and Riordan, (-1, 1) and are zero elsewhere. John F. John F. (over) (over) UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Mathematics-Tables A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functions- A TABULATION OF FOURIER TRANSFORMS OF TRIGONOMET- 2. Legendre functionsRIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables RIC FUNCTIONS AND LEGENDRE POLYNOMIALS by Elmer G. Tables Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's series- Gilbert, Joseph Otterman, and John F. Riordan. Sep 60. 24 p. 3. Fourier's seriesincl. tables, 3 refs. Tables incl. tables, 3 refs. Tables (Rept. no. 36943-15-T) Unclassified report 4. Trigonometry- (Rept. no. 36943-15-T) Unclassified report 4. TrigonometryThis report presents tables of Fourier transforms of functions which Tables This report presents tables of Fourier transforms of functions which Tabes are trigonometric functions or Legendre polynomials on the interval I. Gilbert Elmer., Otterman, are trigonometric functions or Legendre polynomials on the interval, (-1 1) and are zero elsewhere. oseph, and Riordan, (-1, 1) and are zero elsewhere. Jo h n R d John F. John F. (over) (over) UNCLASSIFIED UNCLASSIFIED ++ +

UNCLASSIFIED UNCLASSIFIED UNITERMS UNITERMS Fourier transforms Fourier transforms Trigonometric functions Trigonometric functions Legendre polynomials Legendre polynomials UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UNITERMS UNITERMS Fourier transforms Fourier transforms Trigonometric functions Trigonometric functions Legendre polynomials Legendre polynomials UNCLASSIFIED