36:
908:
725:
1944:
805:
1822:
1316:
1025:
1456:
472:
1075:
1497:
385:
2624:
2391:
2559:
2484:
413:
1396:
1221:
2262:
1734:
1179:
969:
167:
2231:
2199:
2163:
1762:
1254:
1148:
936:
498:
1707:
lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to
237:
2132:
2089:
1116:
661:
307:
2008:
1354:
762:
277:
203:
1976:
2514:
2451:
789:
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2419:
2354:
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1627:
1603:
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622:
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903:{\displaystyle {\mathcal {Z}}:=\left\{\left(\cos ^{2}\theta ,{\tfrac {1}{2}}\sin 2\theta \right)\mid 0<\theta <\pi \right\}}
100:
72:
2686:
1769:
1263:
978:
1403:
421:
79:
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119:
53:
1036:
86:
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1461:
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57:
68:
2732:
2273:
1837:
334:
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390:
17:
1363:
1188:
2727:
2240:
1712:
1157:
945:
145:
2212:
2180:
2144:
1743:
1235:
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917:
477:
558:
in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes
2588:
46:
2747:
2742:
2288:
1556:
93:
2661:
2331:
2015:
1257:
324:
312:
216:
2300:
2111:
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1086:
631:
286:
2525:
2095:
1981:
1327:
735:
250:
176:
1952:
8:
2629:
2587:
copies of the upper half-plane. Yet another space interesting to number theorists is the
2428:
2424:
2269:
1833:
1656:
2499:
2433:
771:
2656:
2568:
2404:
2339:
2319:
2315:
2304:
2292:
2048:
2022:
1688:
1664:
1636:
1612:
1588:
1564:
1536:
1512:
607:
581:
561:
535:
511:
2204:
1659:
can be used to define a distance that is invariant under dilation. In the former case
2703:
2700:
2641:
2277:
2172:
1824:
and logarithmic measure on this ray. In consequence, the upper half-plane becomes a
2396:
2335:
2099:
2011:
1829:
1226:
2681:
208:
2666:
2651:
2489:
2168:
2137:
1857:
1853:
2721:
2671:
2521:
2281:
1849:
720:{\displaystyle \lambda =({\text{diameter of}}\ B)/({\text{diameter of}}\ A)}
2103:
1825:
1939:{\displaystyle {\mathcal {H}}:=\{x+iy\mid y>0;\ x,y\in \mathbb {R} \}.}
1319:
133:
1607:
either intersects the boundary or is parallel to it. In the latter case
2528:
is concerned with the study of certain functions on the direct product
2042:
1030:
555:
2708:
2492:
1555:
in the upper half-plane can be consistently defined as follows: The
35:
1949:
The term arises from a common visualization of the complex number
2019:
1848:
Mathematicians sometimes identify the
Cartesian plane with the
1852:, and then the upper half-plane corresponds to the set of
2698:
2207:"), meaning that it is usually possible to pass between
1766:
can be defined using the correspondence with points on
990:
950:
849:
2598:
2571:
2536:
2502:
2461:
2436:
2407:
2365:
2342:
2318:
of the upper half-plane and the real axis. It is the
2243:
2215:
2183:
2147:
2114:
2071:
2051:
2025:
1984:
1955:
1869:
1772:
1746:
1715:
1691:
1667:
1639:
1615:
1591:
1567:
1539:
1515:
1464:
1406:
1366:
1330:
1266:
1238:
1191:
1160:
1127:
1089:
1039:
981:
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808:
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738:
671:
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610:
584:
564:
538:
514:
480:
424:
393:
337:
289:
253:
219:
179:
148:
1817:{\displaystyle {\bigl \{}(1,y)\mid y>0{\bigr \}}}
1311:{\displaystyle {\bigl \{}(1,y)\mid y>0{\bigr \}}}
1020:{\displaystyle {\bigl (}{\tfrac {1}{2}},0{\bigr )},}
1451:{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
60:. Unsourced material may be challenged and removed.
2618:
2577:
2553:
2508:
2478:
2445:
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2385:
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2193:
2157:
2136:is equally good, but less used by convention. The
2126:
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2002:
1970:
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1816:
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1673:
1645:
1621:
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1310:
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1215:
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1142:
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963:
930:
902:
783:
756:
719:
655:
616:
590:
570:
544:
520:
492:
467:{\displaystyle (x,y)\mapsto (\lambda x,\lambda y)}
466:
407:
379:
301:
271:
231:
197:
161:
2719:
2453:. In this terminology, the upper half-plane is
27:Complex numbers with non-negative imaginary part
1828:. The generic name of this metric space is the
1655:lie on a ray perpendicular to the boundary and
311:instead. Each is an example of two-dimensional
1809:
1775:
1303:
1269:
1009:
984:
2280:. The Poincaré metric provides a hyperbolic
1930:
1880:
1070:{\displaystyle \rho (\theta )=\cos \theta .}
1492:{\displaystyle \rho (\theta )=\cos \theta }
1836:, this model is frequently designated the
940:can be recognized as the circle of radius
1926:
401:
120:Learn how and when to remove this message
2065:axis and thus complex numbers for which
14:
2720:
2045:" corresponds to the region above the
2699:
795:
1507:The distance between any two points
380:{\displaystyle (x,y)\mapsto (x+c,y)}
58:adding citations to reliable sources
29:
2619:{\displaystyle {\mathcal {H}}_{n},}
2386:{\displaystyle {\mathcal {H}}^{n},}
2303:of surfaces with constant negative
2268:It also plays an important role in
2167:(the set of all complex numbers of
2106:. The lower half-plane, defined by
2041:is oriented vertically, the "upper
24:
2602:
2554:{\displaystyle {\mathcal {H}}^{n}}
2540:
2479:{\displaystyle {\mathcal {H}}^{2}}
2465:
2369:
2325:
2246:
2218:
2186:
2171:less than one) is equivalent by a
2150:
1872:
1749:
1718:
1502:
1499:is the reciprocal of that length.
1241:
1163:
923:
811:
318:
151:
25:
2759:
2647:Extended complex upper-half plane
2098:of many functions of interest in
602:Proof: First shift the center of
408:{\displaystyle c\in \mathbb {R} }
1843:
1391:{\displaystyle (1,\tan \theta )}
1216:{\displaystyle (1,\tan \theta )}
327:of the upper half-plane include
34:
2677:Moduli stack of elliptic curves
2257:{\displaystyle {\mathcal {D}}.}
1729:{\displaystyle {\mathcal {Z}}.}
1174:{\displaystyle {\mathcal {Z}},}
964:{\displaystyle {\tfrac {1}{2}}}
162:{\displaystyle {\mathcal {H}},}
45:needs additional citations for
2330:One natural generalization in
2226:{\displaystyle {\mathcal {H}}}
2194:{\displaystyle {\mathcal {H}}}
2158:{\displaystyle {\mathcal {D}}}
1997:
1985:
1792:
1780:
1757:{\displaystyle {\mathcal {Z}}}
1474:
1468:
1385:
1367:
1343:
1331:
1286:
1274:
1249:{\displaystyle {\mathcal {Z}}}
1210:
1192:
1143:{\displaystyle \rho (\theta )}
1137:
1131:
1102:
1090:
1049:
1043:
931:{\displaystyle {\mathcal {Z}}}
751:
739:
714:
700:
692:
678:
647:
635:
493:{\displaystyle \lambda >0.}
461:
443:
440:
437:
425:
374:
356:
353:
350:
338:
266:
254:
192:
180:
13:
1:
2692:
2687:Schwarz–Ahlfors–Pick theorem
2276:provides a way of examining
1832:. In terms of the models of
1322:. Indeed, the diagonal from
7:
2635:
10:
2764:
2395:the maximally symmetric,
2322:of the upper half-plane.
2274:Poincaré half-plane model
1838:Poincaré half-plane model
2301:universal covering space
2628:which is the domain of
2589:Siegel upper half-space
2312:closed upper half-plane
730:and dilate. Then shift
232:{\displaystyle y>0.}
2620:
2579:
2555:
2510:
2480:
2447:
2415:
2387:
2350:
2289:uniformization theorem
2258:
2227:
2195:
2159:
2128:
2127:{\displaystyle y<0}
2085:
2084:{\displaystyle y>0}
2059:
2033:
2004:
1972:
1940:
1818:
1758:
1730:
1699:
1675:
1647:
1623:
1599:
1575:
1557:perpendicular bisector
1547:
1523:
1493:
1452:
1392:
1350:
1312:
1250:
1217:
1175:
1144:
1112:
1111:{\displaystyle (0,0),}
1071:
1021:
965:
932:
904:
785:
758:
721:
657:
656:{\displaystyle (0,0).}
618:
592:
572:
546:
522:
494:
468:
409:
381:
325:affine transformations
303:
302:{\displaystyle y<0}
273:
233:
199:
163:
2738:Differential geometry
2621:
2580:
2556:
2526:Hilbert modular forms
2511:
2481:
2448:
2416:
2388:
2351:
2332:differential geometry
2259:
2228:
2196:
2160:
2129:
2086:
2060:
2034:
2016:Cartesian coordinates
2005:
2003:{\displaystyle (x,y)}
1973:
1941:
1819:
1759:
1731:
1700:
1676:
1648:
1624:
1600:
1576:
1548:
1524:
1494:
1453:
1393:
1351:
1349:{\displaystyle (0,0)}
1313:
1251:
1218:
1176:
1145:
1113:
1072:
1022:
966:
933:
905:
786:
759:
757:{\displaystyle (0,0)}
722:
658:
619:
593:
573:
547:
523:
495:
469:
410:
382:
304:
274:
272:{\displaystyle (x,y)}
245:is the set of points
234:
200:
198:{\displaystyle (x,y)}
171:is the set of points
164:
2630:Siegel modular forms
2596:
2569:
2534:
2500:
2459:
2434:
2405:
2363:
2340:
2241:
2213:
2181:
2145:
2112:
2069:
2049:
2023:
1982:
1971:{\displaystyle x+iy}
1953:
1867:
1770:
1744:
1713:
1689:
1665:
1637:
1613:
1589:
1565:
1559:of the segment from
1537:
1513:
1462:
1404:
1364:
1328:
1264:
1236:
1189:
1158:
1125:
1087:
1037:
979:
946:
918:
806:
772:
736:
669:
632:
608:
582:
562:
536:
512:
478:
422:
391:
335:
287:
251:
217:
177:
146:
54:improve this article
2733:Hyperbolic geometry
2429:sectional curvature
2425:Riemannian manifold
2270:hyperbolic geometry
1834:hyperbolic geometry
1657:logarithmic measure
1400:has squared length
2704:"Upper Half-Plane"
2701:Weisstein, Eric W.
2657:Fundamental domain
2616:
2575:
2551:
2509:{\displaystyle 2.}
2506:
2476:
2446:{\displaystyle -1}
2443:
2411:
2383:
2346:
2305:Gaussian curvature
2278:hyperbolic motions
2254:
2223:
2191:
2155:
2124:
2081:
2055:
2029:
2000:
1968:
1936:
1814:
1754:
1726:
1695:
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1643:
1619:
1595:
1571:
1543:
1519:
1489:
1448:
1388:
1346:
1308:
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1213:
1171:
1140:
1108:
1067:
1017:
999:
961:
959:
928:
900:
858:
796:Inversive geometry
784:{\displaystyle B.}
781:
754:
717:
653:
614:
588:
568:
542:
518:
490:
464:
405:
377:
299:
269:
229:
195:
159:
69:"Upper half-plane"
2642:Cusp neighborhood
2578:{\displaystyle n}
2414:{\displaystyle n}
2349:{\displaystyle n}
2173:conformal mapping
2058:{\displaystyle x}
2032:{\displaystyle y}
1912:
1698:{\displaystyle q}
1674:{\displaystyle p}
1646:{\displaystyle q}
1622:{\displaystyle p}
1598:{\displaystyle q}
1574:{\displaystyle p}
1546:{\displaystyle q}
1522:{\displaystyle p}
998:
958:
857:
766:to the center of
710:
706:
688:
684:
617:{\displaystyle A}
591:{\displaystyle B}
571:{\displaystyle A}
545:{\displaystyle B}
521:{\displaystyle A}
130:
129:
122:
104:
16:(Redirected from
2755:
2728:Complex analysis
2714:
2713:
2627:
2625:
2623:
2622:
2617:
2612:
2611:
2606:
2605:
2586:
2584:
2582:
2581:
2576:
2562:
2560:
2558:
2557:
2552:
2550:
2549:
2544:
2543:
2524:, the theory of
2517:
2515:
2513:
2512:
2507:
2487:
2485:
2483:
2482:
2477:
2475:
2474:
2469:
2468:
2452:
2450:
2449:
2444:
2422:
2420:
2418:
2417:
2412:
2397:simply connected
2394:
2392:
2390:
2389:
2384:
2379:
2378:
2373:
2372:
2355:
2353:
2352:
2347:
2297:upper half-plane
2295:states that the
2265:
2263:
2261:
2260:
2255:
2250:
2249:
2234:
2232:
2230:
2229:
2224:
2222:
2221:
2202:
2200:
2198:
2197:
2192:
2190:
2189:
2166:
2164:
2162:
2161:
2156:
2154:
2153:
2135:
2133:
2131:
2130:
2125:
2100:complex analysis
2090:
2088:
2087:
2082:
2064:
2062:
2061:
2056:
2038:
2036:
2035:
2030:
2009:
2007:
2006:
2001:
1977:
1975:
1974:
1969:
1945:
1943:
1942:
1937:
1929:
1910:
1876:
1875:
1830:hyperbolic plane
1823:
1821:
1820:
1815:
1813:
1812:
1779:
1778:
1765:
1763:
1761:
1760:
1755:
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1722:
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1704:
1702:
1701:
1696:
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1554:
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1528:
1526:
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1520:
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1457:
1455:
1454:
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1422:
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1399:
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1395:
1394:
1389:
1357:
1355:
1353:
1352:
1347:
1317:
1315:
1314:
1309:
1307:
1306:
1273:
1272:
1255:
1253:
1252:
1247:
1245:
1244:
1227:collinear points
1224:
1222:
1220:
1219:
1214:
1182:
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1177:
1172:
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1166:
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1073:
1068:
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1024:
1023:
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1000:
991:
988:
987:
972:
970:
968:
967:
962:
960:
951:
939:
937:
935:
934:
929:
927:
926:
909:
907:
906:
901:
899:
895:
876:
872:
859:
850:
838:
837:
815:
814:
792:
790:
788:
787:
782:
765:
763:
761:
760:
755:
726:
724:
723:
718:
708:
707:
704:
699:
686:
685:
682:
664:
662:
660:
659:
654:
625:
623:
621:
620:
615:
597:
595:
594:
589:
577:
575:
574:
569:
553:
551:
549:
548:
543:
529:
527:
525:
524:
519:
499:
497:
496:
491:
473:
471:
470:
465:
414:
412:
411:
406:
404:
386:
384:
383:
378:
310:
308:
306:
305:
300:
280:
278:
276:
275:
270:
243:lower half-plane
240:
238:
236:
235:
230:
206:
204:
202:
201:
196:
170:
168:
166:
165:
160:
155:
154:
138:upper half-plane
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
2763:
2762:
2758:
2757:
2756:
2754:
2753:
2752:
2718:
2717:
2695:
2682:Riemann surface
2638:
2607:
2601:
2600:
2599:
2597:
2594:
2593:
2591:
2570:
2567:
2566:
2564:
2545:
2539:
2538:
2537:
2535:
2532:
2531:
2529:
2501:
2498:
2497:
2495:
2470:
2464:
2463:
2462:
2460:
2457:
2456:
2454:
2435:
2432:
2431:
2406:
2403:
2402:
2400:
2374:
2368:
2367:
2366:
2364:
2361:
2360:
2358:
2341:
2338:
2337:
2328:
2326:Generalizations
2245:
2244:
2242:
2239:
2238:
2236:
2217:
2216:
2214:
2211:
2210:
2208:
2205:Poincaré metric
2185:
2184:
2182:
2179:
2178:
2176:
2149:
2148:
2146:
2143:
2142:
2140:
2113:
2110:
2109:
2107:
2070:
2067:
2066:
2050:
2047:
2046:
2024:
2021:
2020:
1983:
1980:
1979:
1954:
1951:
1950:
1925:
1871:
1870:
1868:
1865:
1864:
1854:complex numbers
1846:
1808:
1807:
1774:
1773:
1771:
1768:
1767:
1748:
1747:
1745:
1742:
1741:
1739:
1717:
1716:
1714:
1711:
1710:
1708:
1690:
1687:
1686:
1684:
1666:
1663:
1662:
1660:
1638:
1635:
1634:
1632:
1614:
1611:
1610:
1608:
1590:
1587:
1586:
1584:
1566:
1563:
1562:
1560:
1538:
1535:
1534:
1532:
1514:
1511:
1510:
1508:
1505:
1503:Metric geometry
1463:
1460:
1459:
1436:
1432:
1417:
1413:
1405:
1402:
1401:
1365:
1362:
1361:
1359:
1329:
1326:
1325:
1323:
1302:
1301:
1268:
1267:
1265:
1262:
1261:
1240:
1239:
1237:
1234:
1233:
1190:
1187:
1186:
1184:
1162:
1161:
1159:
1156:
1155:
1153:
1126:
1123:
1122:
1120:
1088:
1085:
1084:
1082:
1038:
1035:
1034:
1008:
1007:
989:
983:
982:
980:
977:
976:
974:
949:
947:
944:
943:
941:
922:
921:
919:
916:
915:
913:
848:
833:
829:
828:
824:
823:
819:
810:
809:
807:
804:
803:
798:
773:
770:
769:
767:
737:
734:
733:
731:
703:
695:
681:
670:
667:
666:
633:
630:
629:
627:
609:
606:
605:
603:
583:
580:
579:
563:
560:
559:
537:
534:
533:
531:
513:
510:
509:
507:
479:
476:
475:
423:
420:
419:
400:
392:
389:
388:
336:
333:
332:
321:
319:Affine geometry
288:
285:
284:
282:
252:
249:
248:
246:
218:
215:
214:
212:
209:Cartesian plane
178:
175:
174:
172:
150:
149:
147:
144:
143:
141:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
2761:
2751:
2750:
2745:
2740:
2735:
2730:
2716:
2715:
2694:
2691:
2690:
2689:
2684:
2679:
2674:
2669:
2667:Kleinian group
2664:
2659:
2654:
2652:Fuchsian group
2649:
2644:
2637:
2634:
2615:
2610:
2604:
2574:
2548:
2542:
2505:
2473:
2467:
2442:
2439:
2427:with constant
2410:
2382:
2377:
2371:
2345:
2327:
2324:
2284:on the space.
2253:
2248:
2220:
2188:
2169:absolute value
2152:
2138:open unit disk
2123:
2120:
2117:
2080:
2077:
2074:
2054:
2028:
1999:
1996:
1993:
1990:
1987:
1967:
1964:
1961:
1958:
1947:
1946:
1935:
1932:
1928:
1924:
1921:
1918:
1915:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1874:
1858:imaginary part
1856:with positive
1845:
1842:
1811:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1777:
1751:
1725:
1720:
1694:
1670:
1642:
1618:
1594:
1570:
1542:
1518:
1504:
1501:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1447:
1444:
1439:
1435:
1431:
1428:
1425:
1420:
1416:
1412:
1409:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1345:
1342:
1339:
1336:
1333:
1305:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1271:
1243:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1170:
1165:
1139:
1136:
1133:
1130:
1107:
1104:
1101:
1098:
1095:
1092:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1016:
1011:
1006:
1003:
997:
994:
986:
957:
954:
925:
898:
894:
891:
888:
885:
882:
879:
875:
871:
868:
865:
862:
856:
853:
847:
844:
841:
836:
832:
827:
822:
818:
813:
797:
794:
780:
777:
753:
750:
747:
744:
741:
728:
727:
716:
713:
702:
698:
694:
691:
680:
677:
674:
652:
649:
646:
643:
640:
637:
613:
587:
567:
541:
517:
501:
500:
489:
486:
483:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
416:
403:
399:
396:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
320:
317:
298:
295:
292:
268:
265:
262:
259:
256:
228:
225:
222:
194:
191:
188:
185:
182:
158:
153:
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
2760:
2749:
2748:Modular forms
2746:
2744:
2743:Number theory
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2725:
2723:
2711:
2710:
2705:
2702:
2697:
2696:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2672:Modular group
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2653:
2650:
2648:
2645:
2643:
2640:
2639:
2633:
2631:
2613:
2608:
2590:
2572:
2546:
2527:
2523:
2522:number theory
2518:
2503:
2494:
2491:
2488:since it has
2471:
2440:
2437:
2430:
2426:
2423:-dimensional
2408:
2398:
2380:
2375:
2357:
2343:
2333:
2323:
2321:
2317:
2313:
2308:
2306:
2302:
2298:
2294:
2290:
2285:
2283:
2279:
2275:
2271:
2266:
2251:
2206:
2174:
2170:
2139:
2121:
2118:
2115:
2105:
2104:modular forms
2102:, especially
2101:
2097:
2092:
2078:
2075:
2072:
2052:
2044:
2040:
2026:
2017:
2014:endowed with
2013:
1994:
1991:
1988:
1978:as the point
1965:
1962:
1959:
1956:
1933:
1922:
1919:
1916:
1913:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1877:
1863:
1862:
1861:
1859:
1855:
1851:
1850:complex plane
1844:Complex plane
1841:
1839:
1835:
1831:
1827:
1804:
1801:
1798:
1795:
1789:
1786:
1783:
1738:Distances on
1723:
1692:
1668:
1658:
1640:
1616:
1592:
1568:
1558:
1540:
1516:
1500:
1486:
1483:
1480:
1477:
1471:
1465:
1445:
1442:
1437:
1433:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1382:
1379:
1376:
1373:
1370:
1340:
1337:
1334:
1321:
1298:
1295:
1292:
1289:
1283:
1280:
1277:
1259:
1230:
1228:
1207:
1204:
1201:
1198:
1195:
1168:
1134:
1128:
1105:
1099:
1096:
1093:
1081:
1077:
1064:
1061:
1058:
1055:
1052:
1046:
1040:
1032:
1014:
1004:
1001:
995:
992:
955:
952:
911:
896:
892:
889:
886:
883:
880:
877:
873:
869:
866:
863:
860:
854:
851:
845:
842:
839:
834:
830:
825:
820:
816:
802:
793:
778:
775:
748:
745:
742:
711:
696:
689:
675:
672:
650:
644:
641:
638:
611:
601:
600:
599:
585:
565:
557:
539:
515:
505:
487:
484:
481:
458:
455:
452:
449:
446:
434:
431:
428:
417:
397:
394:
371:
368:
365:
362:
359:
347:
344:
341:
330:
329:
328:
326:
316:
314:
296:
293:
290:
263:
260:
257:
244:
226:
223:
220:
210:
189:
186:
183:
156:
139:
135:
124:
121:
113:
110:February 2010
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
2707:
2519:
2329:
2311:
2309:
2296:
2286:
2272:, where the
2267:
2093:
1948:
1847:
1826:metric space
1506:
1260:of the line
1231:
1080:Proposition:
1079:
1078:
973:centered at
912:
800:
799:
729:
504:Proposition:
503:
502:
322:
242:
137:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2336:hyperbolic
2018:. When the
1320:unit circle
1029:and as the
801:Definition:
705:diameter of
683:diameter of
556:semicircles
134:mathematics
2722:Categories
2693:References
2662:Half-space
2094:It is the
2043:half-plane
1458:, so that
1031:polar plot
665:Then take
418:dilations
313:half-space
80:newspapers
18:Half-plane
2709:MathWorld
2493:dimension
2438:−
2012:the plane
1923:∈
1896:∣
1796:∣
1487:θ
1484:
1472:θ
1466:ρ
1446:θ
1443:
1427:θ
1424:
1383:θ
1380:
1290:∣
1258:inversion
1232:In fact,
1208:θ
1205:
1135:θ
1129:ρ
1062:θ
1059:
1047:θ
1041:ρ
893:π
887:θ
878:∣
870:θ
864:
843:θ
840:
673:λ
482:λ
456:λ
447:λ
441:↦
398:∈
354:↦
2636:See also
2293:surfaces
2626:
2592:
2585:
2565:
2561:
2530:
2516:
2496:
2486:
2455:
2421:
2401:
2393:
2359:
2320:closure
2314:is the
2299:is the
2264:
2237:
2233:
2209:
2201:
2177:
2165:
2141:
2134:
2108:
1764:
1740:
1736:
1709:
1705:
1685:
1681:
1661:
1653:
1633:
1629:
1609:
1605:
1585:
1581:
1561:
1553:
1533:
1529:
1509:
1398:
1360:
1356:
1324:
1318:in the
1256:is the
1223:
1185:
1181:
1154:
1150:
1121:
1118:
1083:
1027:
975:
971:
942:
938:
914:
791:
768:
764:
732:
663:
628:
624:
604:
552:
532:
528:
508:
331:shifts
309:
283:
279:
247:
239:
213:
207:in the
205:
173:
169:
142:
94:scholar
2356:-space
2282:metric
2203:(see "
2096:domain
1911:
709:
687:
136:, the
96:
89:
82:
75:
67:
2316:union
415:, and
281:with
211:with
101:JSTOR
87:books
2490:real
2310:The
2291:for
2287:The
2235:and
2119:<
2076:>
2039:axis
1902:>
1802:>
1683:and
1631:and
1531:and
1296:>
1225:are
1183:and
1033:of
890:<
884:<
530:and
506:Let
485:>
323:The
294:<
241:The
224:>
73:news
2563:of
2520:In
2334:is
2175:to
2010:in
1583:to
1481:cos
1434:sec
1415:tan
1377:tan
1358:to
1202:tan
1152:in
1056:cos
861:sin
831:cos
626:to
578:to
554:be
132:In
56:by
2724::
2706:.
2632:.
2504:2.
2399:,
2307:.
2091:.
1878::=
1860::
1840:.
1229:.
910:.
817::=
598:.
488:0.
474:,
387:,
315:.
227:0.
140:,
2712:.
2614:,
2609:n
2603:H
2573:n
2547:n
2541:H
2472:2
2466:H
2441:1
2409:n
2381:,
2376:n
2370:H
2344:n
2252:.
2247:D
2219:H
2187:H
2151:D
2122:0
2116:y
2079:0
2073:y
2053:x
2027:y
1998:)
1995:y
1992:,
1989:x
1986:(
1966:y
1963:i
1960:+
1957:x
1934:.
1931:}
1927:R
1920:y
1917:,
1914:x
1908:;
1905:0
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1893:y
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1881:{
1873:H
1810:}
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1793:)
1790:y
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1784:1
1781:(
1776:{
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1724:.
1719:Z
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1669:p
1641:q
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1478:=
1475:)
1469:(
1438:2
1430:=
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1374:,
1371:1
1368:(
1344:)
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1335:0
1332:(
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1284:y
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1270:{
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1169:,
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1132:(
1106:,
1103:)
1100:0
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1094:0
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1005:0
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743:0
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679:(
676:=
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648:)
645:0
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612:A
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429:x
426:(
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366:c
363:+
360:x
357:(
351:)
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221:y
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184:x
181:(
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117:(
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108:(
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91:·
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50:.
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