352:
1975:
2217:
827:
1044:
1732:
305:, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
2955:
1267:
296:
3191:
In other words, applying the Abel transform to a 1-dimensional function and then applying the
Fourier transform to that result is the same as applying the Hankel transform to that function. This concept can be extended to higher dimensions.
1721:
502:
2651:
1990:
157:
1595:
649:
885:
2469:
1970:{\displaystyle -{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {F'(y)}{\sqrt {y^{2}-r^{2}}}}\,dy=\int _{r}^{\infty }\int _{y}^{\infty }{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}f'(s)\,dsdy.}
618:
2674:
1485:
1126:
328:
from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as
180:
343:. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
3220:) is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.
2295:
2345:
3073:
2368:
2318:
2978:
1359:
3104:
has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains
1422:
3186:
3102:
3044:
3011:
2531:
2502:
2262:
1387:
1334:
1314:
843:
The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function
409:
2212:{\displaystyle \int _{r}^{\infty }\int _{r}^{s}{\frac {-2y}{\pi {\sqrt {(y^{2}-r^{2})(s^{2}-y^{2})}}}}\,dyf'(s)\,ds=\int _{r}^{\infty }(-1)f'(s)\,ds=f(r).}
1610:
2550:
56:
379:) is represented in gray in this figure. The observer is assumed to be located infinitely far from the origin so that the limits of integration are Β±β.
822:{\displaystyle F(y)=2\int _{0}^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx=2\int _{|y|}^{\infty }f(r)\,{\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}},}
1493:
1039:{\displaystyle F(y,z)=\int _{-\infty }^{\infty }f(\rho ,z)\,dx=2\int _{y}^{\infty }{\frac {f(\rho ,z)\rho \,d\rho }{\sqrt {\rho ^{2}-y^{2}}}},}
3017:. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous
2373:
515:) is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at
1601:
556:
2950:{\displaystyle f(r)=\left\Delta F-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.}
355:
A geometrical interpretation of the Abel transform in two dimensions. An observer (I) looks along a line parallel to the
332:(ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed.
329:
340:
3266:
1262:{\displaystyle F(s)=\int _{-\infty }^{\infty }f(r)\,dx=2\int _{s}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-s^{2}}}},}
42:
often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function
363:
above the origin. What the observer sees is the projection (i.e. the integral) of the circularly symmetric function
867:
is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the
1427:
335:
In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in
291:{\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.}
3285:
3295:
872:
3290:
519: = β, so that the limits of integration are Β±β, and all lines of sight are parallel to the
336:
3150:
3126:
2267:
3014:
2323:
309:
3049:
321:
2350:
2300:
3229:
2963:
1392:
1072:
A particular type of axial symmetry is spherical symmetry. In this case, we have a function
3159:
2981:
2533:
is related to the spatial distribution of terminal, non-tethered monomers of the polymers.
3078:
3020:
2987:
2507:
2478:
2238:
8:
1981:
1364:
1339:
1319:
1299:
497:{\displaystyle F(y)=\int _{-\infty }^{\infty }f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx,}
39:
20:
3262:
3138:
35:
1716:{\displaystyle F'(y)=2y\int _{y}^{\infty }{\frac {f'(r)}{\sqrt {r^{2}-y^{2}}}}\,dr.}
3248:
N. H. Abel, Journal fΓΌr die reine und angewandte
Mathematik, 1, pp. 153β157 (1826).
3146:
2646:{\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)r\,dr}{\sqrt {r^{2}-y^{2}}}}.}
152:{\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)r}{\sqrt {r^{2}-y^{2}}}}\,dr.}
3201:
403:
from the origin. Referring to the figure on the right, the observer (I) will see
320:
from the center of the flame, while the inverse Abel transform gives the local
302:
312:
of cylindrical flames or plumes, the forward Abel transform is the integrated
3279:
2472:
2222:
636:
391:) can be interpreted as the projection of a circularly symmetric function
1590:{\displaystyle F(y)=-2\int _{y}^{\infty }f'(r){\sqrt {r^{2}-y^{2}}}\,dr.}
27:
313:
3129:
of integral operators. For example, in two dimensions, if we define
3120:
351:
524:
2464:{\displaystyle \Delta F\equiv \lim _{\epsilon \rightarrow 0}}
1726:
Now substitute this into the inverse Abel transform formula:
1291:
3115:
3108:
additional terms, each of them corresponding to one of the
1100:
plane will then be circularly symmetric and expressible as
16:
Integral transform used in various branches of mathematics
2297:, where it abruptly changes its value by a finite amount
2471:. Such a situation is encountered in tethered polymers (
613:{\displaystyle dx={\frac {r\,dr}{\sqrt {r^{2}-y^{2}}}}}
399:) along a set of parallel lines of sight at a distance
3195:
3046:, and it reduces to the classical Abel transform when
2223:
Generalization of the Abel transform to discontinuous
3162:
3081:
3052:
3023:
2990:
2966:
2677:
2553:
2510:
2481:
2376:
2353:
2326:
2303:
2270:
2241:
1993:
1735:
1613:
1496:
1430:
1395:
1367:
1342:
1322:
1302:
1129:
888:
652:
559:
412:
183:
59:
3180:
3096:
3067:
3038:
3005:
2972:
2949:
2645:
2525:
2496:
2463:
2362:
2339:
2312:
2289:
2256:
2211:
1969:
1715:
1589:
1479:
1416:
1381:
1353:
1328:
1308:
1261:
1038:
821:
612:
496:
290:
151:
3121:Relationship to the Fourier and Hankel transforms
2668:, the inverse Abel transform is however given by
3277:
2475:) exhibiting a vertical phase separation, where
2387:
3153:for circularly symmetric functions states that
2544:) is under these circumstances again given by:
3256:
346:
2504:stands for the polymer density profile and
3259:The Fourier Transform and its Applications
1480:{\displaystyle v'=r/{\sqrt {r^{2}-y^{2}}}}
1292:Verification of the inverse Abel transform
3116:Relationship to other integral transforms
2605:
2184:
2130:
2106:
1951:
1812:
1703:
1577:
1221:
1175:
998:
946:
781:
774:
724:
575:
484:
247:
174:, the inverse Abel transform is given by
139:
3125:The Abel transform is one member of the
1120:. Carrying out the integration, we have
371:) along the line of sight. The function
350:
3149:operator, then the special case of the
1389:, we can integrate by parts by setting
3278:
1272:which is again, the Abel transform of
383:In two dimensions, the Abel transform
3200:Abel transform can be viewed as the
2664:) drops to zero more quickly than 1/
1049:which is just the Abel transform of
170:) drops to zero more quickly than 1/
3196:Relationship to the Radon transform
1316:is continuously differentiable and
875:, we can take that plane to be the
832:which yields the Abel transform of
13:
3053:
2882:
2853:
2822:
2801:
2749:
2725:
2582:
2444:
2416:
2377:
2354:
2332:
2304:
2282:
2150:
2004:
1847:
1832:
1759:
1650:
1528:
1198:
1158:
1153:
969:
923:
918:
757:
681:
441:
436:
330:algebraic reconstruction technique
316:along a ray with closest distance
222:
88:
19:For summation transformation, see
14:
3307:
2536:The Abel transform of a function
3133:as the Abel transform operator,
1096:. The projection onto, say, the
3242:
3091:
3085:
3033:
3027:
3000:
2994:
2812:
2793:
2763:
2741:
2730:
2711:
2687:
2681:
2599:
2593:
2563:
2557:
2520:
2514:
2491:
2485:
2458:
2455:
2436:
2427:
2408:
2402:
2394:
2251:
2245:
2203:
2197:
2181:
2175:
2164:
2155:
2127:
2121:
2098:
2072:
2069:
2043:
1948:
1942:
1926:
1900:
1897:
1871:
1781:
1775:
1672:
1666:
1628:
1622:
1547:
1541:
1506:
1500:
1411:
1405:
1215:
1209:
1172:
1166:
1139:
1133:
992:
980:
943:
931:
904:
892:
771:
765:
751:
743:
662:
656:
422:
416:
193:
187:
105:
99:
69:
63:
1:
3235:
2290:{\displaystyle y=y_{\Delta }}
3204:of an isotropic 2D function
7:
3223:
2340:{\displaystyle y_{\Delta }}
1984:, the last integral equals
10:
3312:
3068:{\displaystyle \Delta F=0}
873:Without loss of generality
347:Geometrical interpretation
18:
3261:. New York: McGraw-Hill.
1361:drop to zero faster than
523:axis. Realizing that the
337:photofragment-ion imaging
3151:projection-slice theorem
2363:{\displaystyle \Delta F}
2313:{\displaystyle \Delta F}
2235:Consider the case where
3015:Heaviside step function
2973:{\displaystyle \delta }
310:absorption spectroscopy
3257:Bracewell, R. (1965).
3182:
3098:
3069:
3040:
3007:
2974:
2951:
2647:
2527:
2498:
2465:
2364:
2341:
2314:
2291:
2258:
2213:
1971:
1717:
1591:
1481:
1418:
1417:{\displaystyle u=f(r)}
1383:
1355:
1330:
1310:
1263:
1040:
823:
614:
498:
380:
322:absorption coefficient
292:
153:
3230:GPS radio occultation
3183:
3181:{\displaystyle FA=H.}
3099:
3070:
3041:
3008:
2975:
2952:
2648:
2528:
2499:
2466:
2365:
2342:
2315:
2292:
2259:
2214:
1972:
1718:
1592:
1482:
1419:
1384:
1356:
1331:
1311:
1264:
1041:
824:
615:
499:
354:
341:photoelectron imaging
293:
154:
3160:
3145:as the zeroth-order
3097:{\displaystyle F(y)}
3079:
3050:
3039:{\displaystyle F(y)}
3021:
3006:{\displaystyle H(x)}
2988:
2982:Dirac delta function
2964:
2675:
2551:
2526:{\displaystyle f(r)}
2508:
2497:{\displaystyle F(y)}
2479:
2374:
2351:
2324:
2301:
2268:
2264:is discontinuous at
2257:{\displaystyle F(y)}
2239:
1991:
1733:
1611:
1494:
1428:
1393:
1365:
1340:
1320:
1300:
1127:
886:
650:
557:
410:
181:
57:
3286:Integral transforms
2886:
2831:
2586:
2154:
2023:
2008:
1851:
1836:
1763:
1654:
1532:
1382:{\displaystyle 1/r}
1202:
1162:
973:
927:
761:
685:
445:
226:
92:
3178:
3094:
3065:
3036:
3003:
2970:
2947:
2872:
2817:
2643:
2572:
2523:
2494:
2461:
2401:
2360:
2337:
2310:
2287:
2254:
2209:
2140:
2009:
1994:
1967:
1837:
1822:
1749:
1713:
1640:
1587:
1518:
1477:
1414:
1379:
1354:{\displaystyle f'}
1351:
1326:
1306:
1259:
1188:
1145:
1036:
959:
910:
819:
737:
671:
610:
550:, it follows that
494:
428:
381:
288:
212:
149:
78:
40:integral transform
21:summation by parts
3296:Niels Henrik Abel
3139:Fourier transform
3112:discontinuities.
2942:
2941:
2905:
2870:
2846:
2845:
2785:
2772:
2706:
2638:
2637:
2386:
2104:
2101:
1932:
1929:
1810:
1809:
1747:
1701:
1700:
1575:
1475:
1329:{\displaystyle f}
1309:{\displaystyle f}
1254:
1253:
1031:
1030:
814:
813:
718:
608:
607:
478:
283:
282:
245:
210:
137:
136:
36:Niels Henrik Abel
3303:
3291:Image processing
3272:
3249:
3246:
3187:
3185:
3184:
3179:
3147:Hankel transform
3103:
3101:
3100:
3095:
3074:
3072:
3071:
3066:
3045:
3043:
3042:
3037:
3012:
3010:
3009:
3004:
2979:
2977:
2976:
2971:
2956:
2954:
2953:
2948:
2943:
2940:
2939:
2927:
2926:
2917:
2916:
2908:
2906:
2904:
2896:
2888:
2885:
2880:
2871:
2863:
2852:
2848:
2847:
2844:
2843:
2830:
2825:
2816:
2815:
2805:
2804:
2788:
2786:
2778:
2773:
2771:
2770:
2758:
2753:
2752:
2734:
2729:
2728:
2707:
2699:
2652:
2650:
2649:
2644:
2639:
2636:
2635:
2623:
2622:
2613:
2612:
2588:
2585:
2580:
2532:
2530:
2529:
2524:
2503:
2501:
2500:
2495:
2470:
2468:
2467:
2462:
2448:
2447:
2420:
2419:
2400:
2369:
2367:
2366:
2361:
2346:
2344:
2343:
2338:
2336:
2335:
2319:
2317:
2316:
2311:
2296:
2294:
2293:
2288:
2286:
2285:
2263:
2261:
2260:
2255:
2218:
2216:
2215:
2210:
2174:
2153:
2148:
2120:
2105:
2103:
2102:
2097:
2096:
2084:
2083:
2068:
2067:
2055:
2054:
2042:
2036:
2025:
2022:
2017:
2007:
2002:
1982:Fubini's theorem
1976:
1974:
1973:
1968:
1941:
1933:
1931:
1930:
1925:
1924:
1912:
1911:
1896:
1895:
1883:
1882:
1870:
1864:
1853:
1850:
1845:
1835:
1830:
1811:
1808:
1807:
1795:
1794:
1785:
1784:
1774:
1765:
1762:
1757:
1748:
1740:
1722:
1720:
1719:
1714:
1702:
1699:
1698:
1686:
1685:
1676:
1675:
1665:
1656:
1653:
1648:
1621:
1600:Differentiating
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1561:
1560:
1551:
1540:
1531:
1526:
1486:
1484:
1483:
1478:
1476:
1474:
1473:
1461:
1460:
1451:
1449:
1438:
1423:
1421:
1420:
1415:
1388:
1386:
1385:
1380:
1375:
1360:
1358:
1357:
1352:
1350:
1335:
1333:
1332:
1327:
1315:
1313:
1312:
1307:
1268:
1266:
1265:
1260:
1255:
1252:
1251:
1239:
1238:
1229:
1228:
1204:
1201:
1196:
1161:
1156:
1045:
1043:
1042:
1037:
1032:
1029:
1028:
1016:
1015:
1006:
1005:
975:
972:
967:
926:
921:
828:
826:
825:
820:
815:
812:
811:
799:
798:
789:
788:
776:
760:
755:
754:
746:
723:
719:
717:
716:
704:
703:
694:
684:
679:
619:
617:
616:
611:
609:
606:
605:
593:
592:
583:
582:
570:
503:
501:
500:
495:
483:
479:
477:
476:
464:
463:
454:
444:
439:
359:axis a distance
297:
295:
294:
289:
284:
281:
280:
268:
267:
258:
257:
249:
246:
244:
236:
228:
225:
220:
211:
203:
158:
156:
155:
150:
138:
135:
134:
122:
121:
112:
111:
94:
91:
86:
3311:
3310:
3306:
3305:
3304:
3302:
3301:
3300:
3276:
3275:
3269:
3253:
3252:
3247:
3243:
3238:
3226:
3202:Radon transform
3198:
3161:
3158:
3157:
3123:
3118:
3080:
3077:
3076:
3051:
3048:
3047:
3022:
3019:
3018:
2989:
2986:
2985:
2965:
2962:
2961:
2935:
2931:
2922:
2918:
2909:
2907:
2897:
2889:
2887:
2881:
2876:
2862:
2839:
2835:
2826:
2821:
2800:
2796:
2789:
2787:
2777:
2766:
2762:
2754:
2748:
2744:
2733:
2724:
2720:
2698:
2697:
2693:
2676:
2673:
2672:
2631:
2627:
2618:
2614:
2589:
2587:
2581:
2576:
2552:
2549:
2548:
2509:
2506:
2505:
2480:
2477:
2476:
2443:
2439:
2415:
2411:
2390:
2375:
2372:
2371:
2370:are defined by
2352:
2349:
2348:
2331:
2327:
2325:
2322:
2321:
2302:
2299:
2298:
2281:
2277:
2269:
2266:
2265:
2240:
2237:
2236:
2233:
2167:
2149:
2144:
2113:
2092:
2088:
2079:
2075:
2063:
2059:
2050:
2046:
2041:
2037:
2026:
2024:
2018:
2013:
2003:
1998:
1992:
1989:
1988:
1934:
1920:
1916:
1907:
1903:
1891:
1887:
1878:
1874:
1869:
1865:
1854:
1852:
1846:
1841:
1831:
1826:
1803:
1799:
1790:
1786:
1767:
1766:
1764:
1758:
1753:
1739:
1734:
1731:
1730:
1694:
1690:
1681:
1677:
1658:
1657:
1655:
1649:
1644:
1614:
1612:
1609:
1608:
1569:
1565:
1556:
1552:
1550:
1533:
1527:
1522:
1495:
1492:
1491:
1469:
1465:
1456:
1452:
1450:
1445:
1431:
1429:
1426:
1425:
1394:
1391:
1390:
1371:
1366:
1363:
1362:
1343:
1341:
1338:
1337:
1321:
1318:
1317:
1301:
1298:
1297:
1294:
1247:
1243:
1234:
1230:
1205:
1203:
1197:
1192:
1157:
1149:
1128:
1125:
1124:
1024:
1020:
1011:
1007:
976:
974:
968:
963:
922:
914:
887:
884:
883:
879:plane, so that
807:
803:
794:
790:
777:
775:
756:
750:
742:
741:
712:
708:
699:
695:
693:
689:
680:
675:
651:
648:
647:
643:, we may write
601:
597:
588:
584:
571:
569:
558:
555:
554:
472:
468:
459:
455:
453:
449:
440:
432:
411:
408:
407:
349:
276:
272:
263:
259:
250:
248:
237:
229:
227:
221:
216:
202:
182:
179:
178:
130:
126:
117:
113:
95:
93:
87:
82:
58:
55:
54:
24:
17:
12:
11:
5:
3309:
3299:
3298:
3293:
3288:
3274:
3273:
3267:
3251:
3250:
3240:
3239:
3237:
3234:
3233:
3232:
3225:
3222:
3197:
3194:
3189:
3188:
3177:
3174:
3171:
3168:
3165:
3122:
3119:
3117:
3114:
3093:
3090:
3087:
3084:
3064:
3061:
3058:
3055:
3035:
3032:
3029:
3026:
3002:
2999:
2996:
2993:
2969:
2958:
2957:
2946:
2938:
2934:
2930:
2925:
2921:
2915:
2912:
2903:
2900:
2895:
2892:
2884:
2879:
2875:
2869:
2866:
2861:
2858:
2855:
2851:
2842:
2838:
2834:
2829:
2824:
2820:
2814:
2811:
2808:
2803:
2799:
2795:
2792:
2784:
2781:
2776:
2769:
2765:
2761:
2757:
2751:
2747:
2743:
2740:
2737:
2732:
2727:
2723:
2719:
2716:
2713:
2710:
2705:
2702:
2696:
2692:
2689:
2686:
2683:
2680:
2654:
2653:
2642:
2634:
2630:
2626:
2621:
2617:
2611:
2608:
2604:
2601:
2598:
2595:
2592:
2584:
2579:
2575:
2571:
2568:
2565:
2562:
2559:
2556:
2522:
2519:
2516:
2513:
2493:
2490:
2487:
2484:
2460:
2457:
2454:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2426:
2423:
2418:
2414:
2410:
2407:
2404:
2399:
2396:
2393:
2389:
2385:
2382:
2379:
2359:
2356:
2334:
2330:
2309:
2306:
2284:
2280:
2276:
2273:
2253:
2250:
2247:
2244:
2232:
2221:
2220:
2219:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2183:
2180:
2177:
2173:
2170:
2166:
2163:
2160:
2157:
2152:
2147:
2143:
2139:
2136:
2133:
2129:
2126:
2123:
2119:
2116:
2112:
2109:
2100:
2095:
2091:
2087:
2082:
2078:
2074:
2071:
2066:
2062:
2058:
2053:
2049:
2045:
2040:
2035:
2032:
2029:
2021:
2016:
2012:
2006:
2001:
1997:
1978:
1977:
1966:
1963:
1960:
1957:
1954:
1950:
1947:
1944:
1940:
1937:
1928:
1923:
1919:
1915:
1910:
1906:
1902:
1899:
1894:
1890:
1886:
1881:
1877:
1873:
1868:
1863:
1860:
1857:
1849:
1844:
1840:
1834:
1829:
1825:
1821:
1818:
1815:
1806:
1802:
1798:
1793:
1789:
1783:
1780:
1777:
1773:
1770:
1761:
1756:
1752:
1746:
1743:
1738:
1724:
1723:
1712:
1709:
1706:
1697:
1693:
1689:
1684:
1680:
1674:
1671:
1668:
1664:
1661:
1652:
1647:
1643:
1639:
1636:
1633:
1630:
1627:
1624:
1620:
1617:
1598:
1597:
1586:
1583:
1580:
1572:
1568:
1564:
1559:
1555:
1549:
1546:
1543:
1539:
1536:
1530:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1472:
1468:
1464:
1459:
1455:
1448:
1444:
1441:
1437:
1434:
1413:
1410:
1407:
1404:
1401:
1398:
1378:
1374:
1370:
1349:
1346:
1325:
1305:
1293:
1290:
1270:
1269:
1258:
1250:
1246:
1242:
1237:
1233:
1227:
1224:
1220:
1217:
1214:
1211:
1208:
1200:
1195:
1191:
1187:
1184:
1181:
1178:
1174:
1171:
1168:
1165:
1160:
1155:
1152:
1148:
1144:
1141:
1138:
1135:
1132:
1047:
1046:
1035:
1027:
1023:
1019:
1014:
1010:
1004:
1001:
997:
994:
991:
988:
985:
982:
979:
971:
966:
962:
958:
955:
952:
949:
945:
942:
939:
936:
933:
930:
925:
920:
917:
913:
909:
906:
903:
900:
897:
894:
891:
830:
829:
818:
810:
806:
802:
797:
793:
787:
784:
780:
773:
770:
767:
764:
759:
753:
749:
745:
740:
736:
733:
730:
727:
722:
715:
711:
707:
702:
698:
692:
688:
683:
678:
674:
670:
667:
664:
661:
658:
655:
627:> 0. Since
621:
620:
604:
600:
596:
591:
587:
581:
578:
574:
568:
565:
562:
530:is related to
505:
504:
493:
490:
487:
482:
475:
471:
467:
462:
458:
452:
448:
443:
438:
435:
431:
427:
424:
421:
418:
415:
348:
345:
324:at a distance
303:image analysis
299:
298:
287:
279:
275:
271:
266:
262:
256:
253:
243:
240:
235:
232:
224:
219:
215:
209:
206:
201:
198:
195:
192:
189:
186:
162:Assuming that
160:
159:
148:
145:
142:
133:
129:
125:
120:
116:
110:
107:
104:
101:
98:
90:
85:
81:
77:
74:
71:
68:
65:
62:
50:) is given by
32:Abel transform
15:
9:
6:
4:
3:
2:
3308:
3297:
3294:
3292:
3289:
3287:
3284:
3283:
3281:
3270:
3268:0-07-007016-4
3264:
3260:
3255:
3254:
3245:
3241:
3231:
3228:
3227:
3221:
3219:
3215:
3211:
3207:
3203:
3193:
3175:
3172:
3169:
3166:
3163:
3156:
3155:
3154:
3152:
3148:
3144:
3141:operator and
3140:
3136:
3132:
3128:
3113:
3111:
3107:
3088:
3082:
3062:
3059:
3056:
3030:
3024:
3016:
2997:
2991:
2983:
2967:
2944:
2936:
2932:
2928:
2923:
2919:
2913:
2910:
2901:
2898:
2893:
2890:
2877:
2873:
2867:
2864:
2859:
2856:
2849:
2840:
2836:
2832:
2827:
2818:
2809:
2806:
2797:
2790:
2782:
2779:
2774:
2767:
2759:
2755:
2745:
2738:
2735:
2721:
2717:
2714:
2708:
2703:
2700:
2694:
2690:
2684:
2678:
2671:
2670:
2669:
2667:
2663:
2659:
2640:
2632:
2628:
2624:
2619:
2615:
2609:
2606:
2602:
2596:
2590:
2577:
2573:
2569:
2566:
2560:
2554:
2547:
2546:
2545:
2543:
2539:
2534:
2517:
2511:
2488:
2482:
2474:
2473:Polymer brush
2452:
2449:
2440:
2433:
2430:
2424:
2421:
2412:
2405:
2397:
2391:
2383:
2380:
2357:
2328:
2307:
2278:
2274:
2271:
2248:
2242:
2230:
2226:
2206:
2200:
2194:
2191:
2188:
2185:
2178:
2171:
2168:
2161:
2158:
2145:
2141:
2137:
2134:
2131:
2124:
2117:
2114:
2110:
2107:
2093:
2089:
2085:
2080:
2076:
2064:
2060:
2056:
2051:
2047:
2038:
2033:
2030:
2027:
2019:
2014:
2010:
1999:
1995:
1987:
1986:
1985:
1983:
1964:
1961:
1958:
1955:
1952:
1945:
1938:
1935:
1921:
1917:
1913:
1908:
1904:
1892:
1888:
1884:
1879:
1875:
1866:
1861:
1858:
1855:
1842:
1838:
1827:
1823:
1819:
1816:
1813:
1804:
1800:
1796:
1791:
1787:
1778:
1771:
1768:
1754:
1750:
1744:
1741:
1736:
1729:
1728:
1727:
1710:
1707:
1704:
1695:
1691:
1687:
1682:
1678:
1669:
1662:
1659:
1645:
1641:
1637:
1634:
1631:
1625:
1618:
1615:
1607:
1606:
1605:
1603:
1584:
1581:
1578:
1570:
1566:
1562:
1557:
1553:
1544:
1537:
1534:
1523:
1519:
1515:
1512:
1509:
1503:
1497:
1490:
1489:
1488:
1470:
1466:
1462:
1457:
1453:
1446:
1442:
1439:
1435:
1432:
1408:
1402:
1399:
1396:
1376:
1372:
1368:
1347:
1344:
1323:
1303:
1289:
1287:
1283:
1279:
1275:
1256:
1248:
1244:
1240:
1235:
1231:
1225:
1222:
1218:
1212:
1206:
1193:
1189:
1185:
1182:
1179:
1176:
1169:
1163:
1150:
1146:
1142:
1136:
1130:
1123:
1122:
1121:
1119:
1116: +
1115:
1112: =
1111:
1107:
1103:
1099:
1095:
1092: +
1091:
1088: +
1087:
1084: =
1083:
1079:
1075:
1070:
1068:
1064:
1060:
1056:
1052:
1033:
1025:
1021:
1017:
1012:
1008:
1002:
999:
995:
989:
986:
983:
977:
964:
960:
956:
953:
950:
947:
940:
937:
934:
928:
915:
911:
907:
901:
898:
895:
889:
882:
881:
880:
878:
874:
870:
866:
863: +
862:
859: =
858:
854:
850:
846:
841:
839:
835:
816:
808:
804:
800:
795:
791:
785:
782:
778:
768:
762:
747:
738:
734:
731:
728:
725:
720:
713:
709:
705:
700:
696:
690:
686:
676:
672:
668:
665:
659:
653:
646:
645:
644:
642:
638:
637:even function
634:
630:
626:
602:
598:
594:
589:
585:
579:
576:
572:
566:
563:
560:
553:
552:
551:
549:
546: +
545:
542: =
541:
537:
533:
529:
526:
522:
518:
514:
510:
491:
488:
485:
480:
473:
469:
465:
460:
456:
450:
446:
433:
429:
425:
419:
413:
406:
405:
404:
402:
398:
394:
390:
386:
378:
374:
370:
366:
362:
358:
353:
344:
342:
338:
333:
331:
327:
323:
319:
315:
311:
306:
304:
285:
277:
273:
269:
264:
260:
254:
251:
241:
238:
233:
230:
217:
213:
207:
204:
199:
196:
190:
184:
177:
176:
175:
173:
169:
165:
146:
143:
140:
131:
127:
123:
118:
114:
108:
102:
96:
83:
79:
75:
72:
66:
60:
53:
52:
51:
49:
45:
41:
37:
33:
29:
22:
3258:
3244:
3217:
3213:
3209:
3205:
3199:
3190:
3142:
3134:
3130:
3124:
3109:
3105:
2959:
2665:
2661:
2657:
2655:
2541:
2537:
2535:
2234:
2228:
2224:
1979:
1725:
1599:
1295:
1285:
1281:
1277:
1273:
1271:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1071:
1066:
1062:
1058:
1054:
1050:
1048:
876:
868:
864:
860:
856:
852:
848:
844:
842:
837:
833:
831:
640:
632:
628:
624:
622:
547:
543:
539:
535:
531:
527:
520:
516:
512:
508:
506:
400:
396:
392:
388:
384:
382:
376:
372:
368:
364:
360:
356:
334:
325:
317:
307:
300:
171:
167:
163:
161:
47:
43:
34:, named for
31:
25:
2320:. That is,
28:mathematics
3280:Categories
3236:References
314:absorbance
3127:FHA cycle
3054:Δ
2968:δ
2929:−
2883:∞
2874:∫
2868:π
2860:−
2854:Δ
2833:−
2823:Δ
2807:−
2802:Δ
2783:π
2775:−
2750:Δ
2739:−
2726:Δ
2718:−
2709:δ
2656:Assuming
2625:−
2583:∞
2574:∫
2453:ϵ
2445:Δ
2431:−
2425:ϵ
2422:−
2417:Δ
2395:→
2392:ϵ
2384:≡
2378:Δ
2355:Δ
2333:Δ
2305:Δ
2283:Δ
2159:−
2151:∞
2142:∫
2086:−
2057:−
2039:π
2028:−
2011:∫
2005:∞
1996:∫
1914:−
1885:−
1867:π
1856:−
1848:∞
1839:∫
1833:∞
1824:∫
1797:−
1760:∞
1751:∫
1745:π
1737:−
1688:−
1651:∞
1642:∫
1563:−
1529:∞
1520:∫
1513:−
1463:−
1296:Assuming
1241:−
1199:∞
1190:∫
1159:∞
1154:∞
1151:−
1147:∫
1108:), where
1080:), where
1018:−
1009:ρ
1003:ρ
996:ρ
984:ρ
970:∞
961:∫
935:ρ
924:∞
919:∞
916:−
912:∫
855:), where
801:−
758:∞
739:∫
682:∞
673:∫
595:−
442:∞
437:∞
434:−
430:∫
270:−
223:∞
214:∫
208:π
200:−
124:−
89:∞
80:∫
3224:See also
2172:′
2118:′
1939:′
1772:′
1663:′
1619:′
1602:formally
1538:′
1487:to find
1436:′
1348:′
635:) is an
38:, is an
3137:as the
2980:is the
1057:,
851:,
3265:
3212:). As
2960:where
871:axis.
525:radius
507:where
30:, the
3075:. If
1280:) in
1061:) in
3263:ISBN
3013:the
2984:and
2347:and
1424:and
1284:and
1065:and
623:for
534:and
339:and
2388:lim
1980:By
840:).
639:in
538:as
308:In
301:In
26:In
3282::
1604:,
1336:,
1288:.
1098:yz
1069:.
877:yz
3271:.
3218:r
3216:(
3214:f
3210:r
3208:(
3206:f
3176:.
3173:H
3170:=
3167:A
3164:F
3143:H
3135:F
3131:A
3110:n
3106:n
3092:)
3089:y
3086:(
3083:F
3063:0
3060:=
3057:F
3034:)
3031:y
3028:(
3025:F
3001:)
2998:x
2995:(
2992:H
2945:.
2937:2
2933:r
2924:2
2920:y
2914:y
2911:d
2902:y
2899:d
2894:F
2891:d
2878:r
2865:1
2857:F
2850:]
2841:2
2837:r
2828:2
2819:y
2813:)
2810:r
2798:y
2794:(
2791:H
2780:1
2768:2
2764:)
2760:r
2756:/
2746:y
2742:(
2736:1
2731:)
2722:y
2715:r
2712:(
2704:2
2701:1
2695:[
2691:=
2688:)
2685:r
2682:(
2679:f
2666:r
2662:r
2660:(
2658:f
2641:.
2633:2
2629:y
2620:2
2616:r
2610:r
2607:d
2603:r
2600:)
2597:r
2594:(
2591:f
2578:y
2570:2
2567:=
2564:)
2561:y
2558:(
2555:F
2542:r
2540:(
2538:f
2521:)
2518:r
2515:(
2512:f
2492:)
2489:y
2486:(
2483:F
2459:]
2456:)
2450:+
2441:y
2437:(
2434:F
2428:)
2413:y
2409:(
2406:F
2403:[
2398:0
2381:F
2358:F
2329:y
2308:F
2279:y
2275:=
2272:y
2252:)
2249:y
2246:(
2243:F
2231:)
2229:y
2227:(
2225:F
2207:.
2204:)
2201:r
2198:(
2195:f
2192:=
2189:s
2186:d
2182:)
2179:s
2176:(
2169:f
2165:)
2162:1
2156:(
2146:r
2138:=
2135:s
2132:d
2128:)
2125:s
2122:(
2115:f
2111:y
2108:d
2099:)
2094:2
2090:y
2081:2
2077:s
2073:(
2070:)
2065:2
2061:r
2052:2
2048:y
2044:(
2034:y
2031:2
2020:s
2015:r
2000:r
1965:.
1962:y
1959:d
1956:s
1953:d
1949:)
1946:s
1943:(
1936:f
1927:)
1922:2
1918:y
1909:2
1905:s
1901:(
1898:)
1893:2
1889:r
1880:2
1876:y
1872:(
1862:y
1859:2
1843:y
1828:r
1820:=
1817:y
1814:d
1805:2
1801:r
1792:2
1788:y
1782:)
1779:y
1776:(
1769:F
1755:r
1742:1
1711:.
1708:r
1705:d
1696:2
1692:y
1683:2
1679:r
1673:)
1670:r
1667:(
1660:f
1646:y
1638:y
1635:2
1632:=
1629:)
1626:y
1623:(
1616:F
1585:.
1582:r
1579:d
1571:2
1567:y
1558:2
1554:r
1548:)
1545:r
1542:(
1535:f
1524:y
1516:2
1510:=
1507:)
1504:y
1501:(
1498:F
1471:2
1467:y
1458:2
1454:r
1447:/
1443:r
1440:=
1433:v
1412:)
1409:r
1406:(
1403:f
1400:=
1397:u
1377:r
1373:/
1369:1
1345:f
1324:f
1304:f
1286:s
1282:r
1278:r
1276:(
1274:f
1257:,
1249:2
1245:s
1236:2
1232:r
1226:r
1223:d
1219:r
1216:)
1213:r
1210:(
1207:f
1194:s
1186:2
1183:=
1180:x
1177:d
1173:)
1170:r
1167:(
1164:f
1143:=
1140:)
1137:s
1134:(
1131:F
1118:z
1114:y
1110:s
1106:s
1104:(
1102:F
1094:z
1090:y
1086:x
1082:r
1078:r
1076:(
1074:f
1067:y
1063:Ο
1059:z
1055:Ο
1053:(
1051:f
1034:,
1026:2
1022:y
1013:2
1000:d
993:)
990:z
987:,
981:(
978:f
965:y
957:2
954:=
951:x
948:d
944:)
941:z
938:,
932:(
929:f
908:=
905:)
902:z
899:,
896:y
893:(
890:F
869:z
865:y
861:x
857:Ο
853:z
849:Ο
847:(
845:f
838:r
836:(
834:f
817:,
809:2
805:y
796:2
792:r
786:r
783:d
779:r
772:)
769:r
766:(
763:f
752:|
748:y
744:|
735:2
732:=
729:x
726:d
721:)
714:2
710:y
706:+
701:2
697:x
691:(
687:f
677:0
669:2
666:=
663:)
660:y
657:(
654:F
641:x
633:r
631:(
629:f
625:x
603:2
599:y
590:2
586:r
580:r
577:d
573:r
567:=
564:x
561:d
548:y
544:x
540:r
536:y
532:x
528:r
521:x
517:x
513:r
511:(
509:f
492:,
489:x
486:d
481:)
474:2
470:y
466:+
461:2
457:x
451:(
447:f
426:=
423:)
420:y
417:(
414:F
401:y
397:r
395:(
393:f
389:y
387:(
385:F
377:r
375:(
373:f
369:r
367:(
365:f
361:y
357:x
326:r
318:y
286:.
278:2
274:r
265:2
261:y
255:y
252:d
242:y
239:d
234:F
231:d
218:r
205:1
197:=
194:)
191:r
188:(
185:f
172:r
168:r
166:(
164:f
147:.
144:r
141:d
132:2
128:y
119:2
115:r
109:r
106:)
103:r
100:(
97:f
84:y
76:2
73:=
70:)
67:y
64:(
61:F
48:r
46:(
44:f
23:.
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