38:: A demonstration of the Euler formula in the section differential equations proof (e^{ix}'=ie^{ix}). I think this demonstration is more direct and intuitive. Observation at the beginning of proofs, since problems with rigor as definition of the e^{ix} used is not cited in some proofs. Added a definition by limit of e^{z} as lim(1+z/n)^n. (not on right now its all under discussion)
2328:
2399:
In fact the finite sums at the right converge to the infinite sum so their diference can be made as smaal as one wants and each term on the initial sum converges to the corresponding term at the end so a finite sum of the terms at the initial sequence can be made as close as one wants to the partial
29:: One paragraph near the end talking about the harmonic motion. I was impressed no one did that before, since it is a very important physics concept. Also pointed trigonometric functions are projections of the circular movement. And explained the animation on the side. Added table for tangents.
1967:
2141:
1768:
2147:
2970:
538:
Alhfors "Complex
Analysis" (1953), Robert B. Ash "Complex Variables"(1971), Anthony B. Holland "Complex function Theory" (1980), Greene/ Krantz "Function Theory of One Complex Variable"(2002), T. Gamelin "Complex Analysis"(2001)
1631:
624:
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One exercise would be to show they are all equivalent, and more so to prove directly the equivalence of each pair (15 of them). There would be also variations if you restricted to real, imaginary, or complex numbers.
2395:
2682:
1789:
1309:
are not equivalent as one would expect and actually there are functions that satisfy the second that are discontinuous everywhere, I think this depends on the axiom of choice but I am not sure.
1973:
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533:
1640:
2323:{\displaystyle =\lim _{n\rightarrow \infty }\sum _{k=0}^{n}(1)\left(1-{\frac {1}{n}}\right)\left(1-{\frac {2}{n}}\right)...\left(1-{\frac {k-1}{n}}\right){\frac {x^{k}}{k!}}\ }
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74:: Added the "If the angle between the sides is right it reduces to the Pythagorean theorem" to make the citation of the cosine law more understandable.
1419:
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349:
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1962:{\displaystyle e^{x}=\lim _{n\rightarrow \infty }\sum _{k=0}^{n}{\frac {(n)(n-1)(n-2)...(n-(k-1))}{k!}}{\frac {x^{k}}{n^{k}}}\ }
147:
197:
129:
2136:{\displaystyle =\lim _{n\rightarrow \infty }\sum _{k=0}^{n}{\frac {(n)(n-1)(n-2)...(n-(k-1))}{n^{k}}}{\frac {x^{k}}{k!}}\ }
32:
To do: The circular movement explains the derivatives of the sine and cosine very nicely, I wonder were that could fit.
2815:
455:
138:
take out POV in the section generalization in "a 'very' important generalization" and "a 'natural' generalization".
50:
To do: Add a picture showing the rectangle and the half right triangle.(already done by someone else, Thank you!)
1763:{\displaystyle e^{x}=\lim _{n\rightarrow \infty }\sum _{k=0}^{n}{\frac {n!}{k!\,(n-k)!}}{\frac {x^{k}}{n^{k}}}\ }
114:: Explained the effect of each term on the graph. Expanded the maximum/minimum analysis for the bivariate case.
221:
239:
47:(circle-triangle): Added a geometrical proof of the converse, I think it makes the converse more intuitive.
26:
257:
176:, already found that the power circle proofs can be simplified(the French one I think its too simplified)
117:
153:
1249:
844:
92:: Added that the diagonal crosses at he midpoints, are equal, and can be calculated using Pythagoras.
629:
E. Townsend "Functions of a complex variable" 1915, Feynman "Lectures on
Physics" Algebra chapter.
71:
1206:
1119:
639:
949:
Definition (2) avoids the problem of showing that such function exists by using Euler's formula.
260:
could be explained by multiple application of trig identities(not rigorous but more insightful).
248:
geometrical interpretation (as rotation and stretching) rules! but need to emphasize that more?
111:
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add that every non-isosceles triangle has two different versions(reflected, non reflected)?
245:
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150:
changed order of relationships, and excluded i and -i from the power series of arctan.
123:
44:
2965:{\displaystyle \cos(x)+i\sin(x)=\lim _{n\rightarrow \infty }(1+i{\frac {x}{n}})^{n}\,}
2980:
1626:{\displaystyle e^{x}=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\ }
619:{\displaystyle e^{z}=\lim _{n\rightarrow \infty }\left(1+{\frac {z}{n}}\right)^{n}\,}
2801:{\displaystyle \cos(x)+i\sin(x)=(\cos({\frac {x}{n}})+i\sin({\frac {x}{n}}))^{n}\,}
126:: Changed wording at the lead, some details on principal root and rational powers.
59:
68:: Added 2R = \frac{abc} {2A} to make the equation more understandable and useful.
251:
120:: Added a_n=a_m+(n-m)*r there a nice interpretation for that(add it somewhere?)
1540:{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{\frac {n!}{k!\,(n-k)!}}a^{n-k}b^{k}\ }
191:
179:
173:
159:
135:
2558:{\displaystyle (\cos(x)+i\sin(x))(\cos(y)+i\sin(y))=\cos(x+y)+i\sin(x+y)\ \,}
800:
as the line integral from 1 to z of 1/z it is needed to choose the "branch".
86:: Added that the ratio c/d is always the same so the definition makes sense.
2810:
When n is big x/n is small so cos(x/n) is almost 1 and sin(x/n) almost x/n.
101:
89:
435:{\displaystyle e^{z}=1+z+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots \,}
65:
80:: Completed the steps on the demonstration using difference of squares.
227:
185:
141:
108:
and fails when parallel sides are equal(explaining why geometrically).
2390:{\displaystyle \rightarrow \sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}\ }
254:
article is much better in French with nice geometric interpretations.
203:
98:: Added a link to Visual complex Analysis a book by Tristan Needham.
62:: Tried a clearer wording for the lead and calculation parts(done?)
1312:
I think this is the best way to introduce the real exponential.
230:
make subsections, change second paragraph of calculus section.
2677:{\displaystyle (\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)\ \,}
340:
and definition (1) and (2) below are by far the most common.
276:
Most of the literature I saw on Euler's formula define first
41:
To do: Don't know what to make of some of the other proofs.
209:
83:
144:
changed second paragraph of calculus section (very rough)
2686:
by induction or by multiple application of the last one.
633:(4) As the unique solution of the differential equation:
132:
added relation to absolute property of complex numbers.
1007:
Definitions (excepting (2)) can also be modified to get
445:
Curtiss(1978), Polya(1974), Courant(1965), Rudin(1966).
224:
give neat demonstration of tangent half angle formula?
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Lets divide the definition of exponential in cases
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gets large, the early terms of the summation (where
1113:
but I would guess (1),(5) are the more common ones.
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952:All definitions above can be modified to get just
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2858:{\displaystyle \approx (1+i{\frac {x}{n}})^{n}\,}
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1981:
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567:
2974:To make it fully rigorous its kind of painful.
528:{\displaystyle e^{z}=e^{x}(\cos(y)+i\sin(y))\,}
1116:A modification of (6) is: the unique function
182:proven equivalent to the parallel postulate?
104:: Area formula based on its sides relates to
8:
1064:I didn't do any research on definitions of
764:Hardy "Course of Pure mathematics" (1908)
725:Lars V. Ahlfors "Complex analysis" (1966).
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212:correct the Indian power series entry.
17:My main interest is in math articles.
2987:Wikipedians interested in mathematics
7:
814:First I will add another definition
218:give the geometrical demonstration.
2924:
2567:by trigonometric identities, then
2404:2B + (trig identities) implies 3B
2357:
2165:
1991:
1817:
1668:
1582:
577:
206:add section of circles on nature.
14:
1302:{\displaystyle f(x+y)=f(x)f(y)\,}
897:{\displaystyle f(x+y)=f(x)f(y)\,}
200:some simplification is needed?
156:details for the SSA ambiguity.
148:Inverse trigonometric functions
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807:My opinions on Euler's formula
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198:Brahmagupta-Fibonacci identity
130:Brahmagupta-Fibonacci identity
1:
2865:for big n. So we should have
1239:{\displaystyle f(x)=a^{x}\,}
1152:{\displaystyle f(x)=a^{x}\,}
817:(6) Unique complex function
679:{\displaystyle f'(z)=f(z)\,}
449:(2) I guess deceptively as
242:article has many problems.
3003:
343:(1) As the Taylor series:
272:Research in the literature
1193:{\displaystyle f'(0)=1\,}
938:{\displaystyle f'(0)=1\,}
793:{\displaystyle \log(z)\,}
755:{\displaystyle \log(z)\,}
976:{\displaystyle e^{ix}\,}
715:{\displaystyle f(0)=1\,}
333:{\displaystyle z=x+yi\,}
222:Trigonometric identities
1085:{\displaystyle e^{x}\,}
1028:{\displaystyle e^{x}\,}
806:
297:{\displaystyle e^{z}\,}
266:
240:Uniform circular motion
162:same details as above.
27:Trigonometric functions
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2190:
2137:
2016:
1963:
1842:
1764:
1693:
1627:
1541:
1468:
1402:
1380:
1355:
1354:{\displaystyle z=x+yi}
1303:
1240:
1194:
1153:
1107:
1086:
1050:
1029:
998:
977:
939:
898:
832:
794:
756:
729:(5) By first defining
716:
680:
620:
529:
436:
334:
298:
188:needs demonstrations.
118:Arithmetic Progression
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335:
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154:Congruence (geometry)
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821:
771:
733:
690:
640:
550:
456:
350:
308:
280:
2400:sum at the right.
1363:(B) Pure imaginary
1106:{\displaystyle x\,}
1049:{\displaystyle x\,}
997:{\displaystyle x\,}
831:{\displaystyle f\,}
258:de Moivre's formula
72:Pythagorean theorem
2962:
2928:
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1379:{\displaystyle yi}
1376:
1351:
1320:(in construction)
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1236:
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1103:
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994:
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828:
811:(in construction)
790:
752:
712:
676:
616:
581:
543:(3) As the limit:
525:
432:
330:
294:
112:Quadratic function
22:Main contributions
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1980:
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1806:
1759:
1755:
1731:
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1622:
1607:
1571:
1536:
1506:
1401:{\displaystyle x}
767:Obs. By defining
602:
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1462:
1444:
1443:
1411:(3) implies (1)
1407:
1405:
1404:
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2407:you can prove
2403:
2374:
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2333:
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2307:
2297:
2274:
2265:
2261:
2230:
2226:
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1471:
1435:
1418:
1417:
1410:
1390:
1389:
1365:
1364:
1328:
1327:
1318:
1316:The many proofs
1248:
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1205:
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905:
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819:
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415:
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390:
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353:
348:
347:
306:
305:
283:
278:
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269:
267:Euler's Formula
263:
246:Complex numbers
233:
216:Heron's formula
165:
106:Heron's formula
96:Complex numbers
78:Heron's formula
53:
36:Euler's formula
20:
16:
12:
11:
5:
3000:
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2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2851:
2847:
2841:
2838:
2833:
2830:
2827:
2824:
2821:
2794:
2790:
2786:
2781:
2778:
2773:
2770:
2767:
2764:
2761:
2758:
2753:
2750:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2622:
2618:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2380:
2377:
2371:
2367:
2359:
2354:
2351:
2348:
2344:
2340:
2313:
2310:
2304:
2300:
2293:
2287:
2283:
2280:
2277:
2271:
2268:
2264:
2260:
2257:
2254:
2250:
2244:
2241:
2236:
2233:
2229:
2224:
2218:
2215:
2210:
2207:
2203:
2199:
2196:
2193:
2188:
2183:
2180:
2177:
2173:
2167:
2164:
2161:
2157:
2153:
2126:
2123:
2117:
2113:
2103:
2099:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2014:
2009:
2006:
2003:
1999:
1993:
1990:
1987:
1983:
1979:
1951:
1947:
1941:
1937:
1928:
1925:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1840:
1835:
1832:
1829:
1825:
1819:
1816:
1813:
1809:
1805:
1800:
1796:
1752:
1748:
1742:
1738:
1729:
1726:
1723:
1720:
1717:
1714:
1710:
1707:
1702:
1699:
1691:
1686:
1683:
1680:
1676:
1670:
1667:
1664:
1660:
1656:
1651:
1647:
1617:
1612:
1606:
1603:
1598:
1595:
1591:
1584:
1581:
1578:
1574:
1570:
1565:
1561:
1549:combined with
1531:
1527:
1521:
1518:
1515:
1511:
1504:
1501:
1498:
1495:
1492:
1489:
1485:
1482:
1477:
1474:
1466:
1461:
1458:
1455:
1451:
1447:
1442:
1438:
1434:
1431:
1428:
1425:
1397:
1375:
1372:
1350:
1347:
1344:
1341:
1338:
1335:
1317:
1314:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1232:
1228:
1224:
1221:
1218:
1215:
1212:
1188:
1185:
1182:
1179:
1176:
1172:
1169:
1145:
1141:
1137:
1134:
1131:
1128:
1125:
1101:
1078:
1074:
1044:
1021:
1017:
992:
969:
966:
962:
946:
945:
933:
930:
927:
924:
921:
917:
914:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
826:
808:
805:
788:
785:
782:
779:
776:
750:
747:
744:
741:
738:
723:
722:
710:
707:
704:
701:
698:
695:
674:
671:
668:
665:
662:
659:
656:
653:
649:
646:
627:
626:
612:
607:
601:
598:
593:
590:
586:
579:
576:
573:
569:
565:
560:
556:
536:
535:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
484:
479:
475:
471:
466:
462:
443:
442:
430:
427:
421:
418:
412:
408:
402:
396:
393:
387:
383:
377:
374:
371:
368:
365:
360:
356:
328:
325:
322:
319:
316:
313:
290:
286:
273:
270:
268:
265:
124:Exponentiation
45:Thales theorem
13:
10:
9:
6:
4:
3:
2:
2999:
2988:
2985:
2984:
2982:
2975:
2972:
2956:
2946:
2943:
2938:
2935:
2932:
2918:
2910:
2904:
2898:
2895:
2892:
2889:
2883:
2877:
2874:
2866:
2849:
2839:
2836:
2831:
2828:
2825:
2819:
2811:
2808:
2792:
2779:
2776:
2768:
2765:
2762:
2759:
2751:
2748:
2740:
2737:
2731:
2725:
2719:
2716:
2713:
2710:
2704:
2698:
2695:
2687:
2684:
2664:
2661:
2655:
2652:
2649:
2646:
2640:
2637:
2631:
2628:
2625:
2620:
2609:
2603:
2600:
2597:
2594:
2588:
2582:
2579:
2568:
2565:
2545:
2542:
2539:
2533:
2530:
2527:
2524:
2518:
2515:
2512:
2506:
2503:
2500:
2491:
2485:
2482:
2479:
2476:
2470:
2464:
2461:
2449:
2443:
2440:
2437:
2434:
2428:
2422:
2419:
2408:
2405:
2401:
2397:
2378:
2375:
2369:
2365:
2352:
2349:
2346:
2342:
2330:
2311:
2308:
2302:
2298:
2291:
2285:
2281:
2278:
2275:
2269:
2266:
2262:
2258:
2255:
2252:
2248:
2242:
2239:
2234:
2231:
2227:
2222:
2216:
2213:
2208:
2205:
2201:
2194:
2186:
2181:
2178:
2175:
2171:
2159:
2151:
2143:
2124:
2121:
2115:
2111:
2101:
2097:
2086:
2083:
2080:
2074:
2071:
2065:
2062:
2059:
2053:
2050:
2047:
2038:
2035:
2032:
2023:
2012:
2007:
2004:
2001:
1997:
1985:
1977:
1969:
1949:
1945:
1939:
1935:
1926:
1923:
1912:
1909:
1906:
1900:
1897:
1891:
1888:
1885:
1879:
1876:
1873:
1864:
1861:
1858:
1849:
1838:
1833:
1830:
1827:
1823:
1811:
1803:
1798:
1794:
1785:
1783:
1779:
1775:
1770:
1750:
1746:
1740:
1736:
1727:
1721:
1718:
1715:
1708:
1705:
1700:
1697:
1689:
1684:
1681:
1678:
1674:
1662:
1654:
1649:
1645:
1636:
1633:
1615:
1610:
1604:
1601:
1596:
1593:
1589:
1576:
1568:
1563:
1559:
1550:
1547:
1529:
1525:
1519:
1516:
1513:
1509:
1502:
1496:
1493:
1490:
1483:
1480:
1475:
1472:
1464:
1459:
1456:
1453:
1449:
1445:
1440:
1432:
1429:
1426:
1415:
1412:
1408:
1395:
1386:
1373:
1370:
1361:
1348:
1345:
1342:
1339:
1336:
1333:
1324:
1321:
1315:
1313:
1310:
1292:
1286:
1280:
1274:
1271:
1265:
1262:
1259:
1253:
1230:
1226:
1222:
1216:
1210:
1201:
1186:
1183:
1177:
1170:
1167:
1143:
1139:
1135:
1129:
1123:
1114:
1099:
1076:
1072:
1061:
1057:
1042:
1019:
1015:
1005:
990:
967:
964:
960:
950:
931:
928:
922:
915:
912:
887:
881:
875:
869:
866:
860:
857:
854:
848:
841:
840:
839:
824:
815:
812:
804:
801:
783:
777:
774:
765:
762:
745:
739:
736:
726:
708:
705:
699:
693:
669:
663:
660:
654:
647:
644:
636:
635:
634:
630:
610:
605:
599:
596:
591:
588:
584:
571:
563:
558:
554:
546:
545:
544:
540:
515:
509:
506:
503:
500:
494:
488:
485:
477:
473:
469:
464:
460:
452:
451:
450:
446:
428:
425:
419:
416:
410:
406:
400:
394:
391:
385:
381:
375:
372:
369:
366:
363:
358:
354:
346:
345:
344:
341:
326:
323:
320:
317:
314:
311:
288:
284:
271:
264:
261:
259:
255:
253:
249:
247:
243:
241:
237:
236:
231:
229:
225:
223:
219:
217:
213:
211:
207:
205:
201:
199:
195:
193:
189:
187:
183:
181:
177:
175:
170:
168:
163:
161:
157:
155:
151:
149:
145:
143:
139:
137:
133:
131:
127:
125:
121:
119:
115:
113:
109:
107:
103:
99:
97:
93:
91:
87:
85:
81:
79:
75:
73:
69:
67:
63:
61:
57:
56:
55:Minor changes
51:
48:
46:
42:
39:
37:
33:
30:
28:
24:
23:
18:
2973:
2867:
2812:
2809:
2688:
2685:
2569:
2566:
2409:
2406:
2402:
2398:
2331:
2144:
1970:
1786:
1781:
1777:
1773:
1771:
1637:
1634:
1551:
1548:
1416:
1413:
1409:
1387:
1362:
1326:(A) Complex
1325:
1322:
1319:
1311:
1202:
1115:
1062:
1058:
1006:
951:
947:
816:
813:
810:
802:
766:
763:
727:
724:
631:
628:
541:
537:
447:
444:
342:
304:for complex
275:
262:
256:
250:
244:
238:
234:
232:
226:
220:
214:
208:
202:
196:
190:
184:
178:
171:
166:
164:
158:
152:
146:
140:
134:
128:
122:
116:
110:
100:
94:
88:
82:
76:
70:
64:
60:Golden ratio
58:
54:
52:
49:
43:
40:
34:
31:
25:
21:
19:
15:
1414:using that
1203:Conditions
252:Determinant
235:Perceptions
172:Revise the
160:Cosine law
1159:such that
838:such that
192:Congruence
180:Pythagoras
174:Cosine law
136:Derivative
2925:∞
2922:→
2899:
2878:
2820:≈
2769:
2741:
2720:
2699:
2656:
2632:
2604:
2583:
2534:
2507:
2486:
2465:
2444:
2423:
2358:∞
2343:∑
2339:→
2279:−
2270:−
2235:−
2209:−
2172:∑
2166:∞
2163:→
2084:−
2075:−
2051:−
2036:−
1998:∑
1992:∞
1989:→
1910:−
1901:−
1877:−
1862:−
1824:∑
1818:∞
1815:→
1784:) become
1719:−
1675:∑
1669:∞
1666:→
1583:∞
1580:→
1517:−
1494:−
1450:∑
1388:(C) Real
1092:for real
1035:for real
983:for real
778:
740:
578:∞
575:→
510:
489:
429:⋯
102:Trapezium
90:Rectangle
2981:Category
1780:<<
1171:′
916:′
648:′
66:Sine law
228:Tangent
186:Pyramid
142:Tangent
2671:
2552:
2385:
2318:
2131:
1957:
1758:
1621:
1535:
204:Circle
686:with
167:To do
1246:and
904:and
2915:lim
2896:sin
2875:cos
2766:sin
2738:cos
2717:sin
2696:cos
2653:sin
2629:cos
2601:sin
2580:cos
2531:sin
2504:cos
2483:sin
2462:cos
2441:sin
2420:cos
2156:lim
1982:lim
1808:lim
1772:as
1659:lim
1635:is
1573:lim
1004:.
775:log
737:log
568:lim
507:sin
486:cos
2983::
1200:.
1056:.
210:Pi
169::
84:Pi
2957:n
2953:)
2947:n
2944:x
2939:i
2936:+
2933:1
2930:(
2919:n
2911:=
2908:)
2905:x
2902:(
2893:i
2890:+
2887:)
2884:x
2881:(
2850:n
2846:)
2840:n
2837:x
2832:i
2829:+
2826:1
2823:(
2793:n
2789:)
2785:)
2780:n
2777:x
2772:(
2763:i
2760:+
2757:)
2752:n
2749:x
2744:(
2735:(
2732:=
2729:)
2726:x
2723:(
2714:i
2711:+
2708:)
2705:x
2702:(
2668:)
2665:x
2662:n
2659:(
2650:i
2647:+
2644:)
2641:x
2638:n
2635:(
2626:=
2621:n
2617:)
2613:)
2610:x
2607:(
2598:i
2595:+
2592:)
2589:x
2586:(
2577:(
2549:)
2546:y
2543:+
2540:x
2537:(
2528:i
2525:+
2522:)
2519:y
2516:+
2513:x
2510:(
2501:=
2498:)
2495:)
2492:y
2489:(
2480:i
2477:+
2474:)
2471:y
2468:(
2459:(
2456:)
2453:)
2450:x
2447:(
2438:i
2435:+
2432:)
2429:x
2426:(
2417:(
2379:!
2376:k
2370:k
2366:x
2353:0
2350:=
2347:k
2312:!
2309:k
2303:k
2299:x
2292:)
2286:n
2282:1
2276:k
2267:1
2263:(
2259:.
2256:.
2253:.
2249:)
2243:n
2240:2
2232:1
2228:(
2223:)
2217:n
2214:1
2206:1
2202:(
2198:)
2195:1
2192:(
2187:n
2182:0
2179:=
2176:k
2160:n
2152:=
2125:!
2122:k
2116:k
2112:x
2102:k
2098:n
2093:)
2090:)
2087:1
2081:k
2078:(
2072:n
2069:(
2066:.
2063:.
2060:.
2057:)
2054:2
2048:n
2045:(
2042:)
2039:1
2033:n
2030:(
2027:)
2024:n
2021:(
2013:n
2008:0
2005:=
2002:k
1986:n
1978:=
1950:k
1946:n
1940:k
1936:x
1927:!
1924:k
1919:)
1916:)
1913:1
1907:k
1904:(
1898:n
1895:(
1892:.
1889:.
1886:.
1883:)
1880:2
1874:n
1871:(
1868:)
1865:1
1859:n
1856:(
1853:)
1850:n
1847:(
1839:n
1834:0
1831:=
1828:k
1812:n
1804:=
1799:x
1795:e
1782:n
1778:k
1774:n
1751:k
1747:n
1741:k
1737:x
1728:!
1725:)
1722:k
1716:n
1713:(
1709:!
1706:k
1701:!
1698:n
1690:n
1685:0
1682:=
1679:k
1663:n
1655:=
1650:x
1646:e
1616:n
1611:)
1605:n
1602:x
1597:+
1594:1
1590:(
1577:n
1569:=
1564:x
1560:e
1530:k
1526:b
1520:k
1514:n
1510:a
1503:!
1500:)
1497:k
1491:n
1488:(
1484:!
1481:k
1476:!
1473:n
1465:n
1460:0
1457:=
1454:k
1446:=
1441:n
1437:)
1433:b
1430:+
1427:a
1424:(
1396:x
1374:i
1371:y
1349:i
1346:y
1343:+
1340:x
1337:=
1334:z
1296:)
1293:y
1290:(
1287:f
1284:)
1281:x
1278:(
1275:f
1272:=
1269:)
1266:y
1263:+
1260:x
1257:(
1254:f
1231:x
1227:a
1223:=
1220:)
1217:x
1214:(
1211:f
1187:1
1184:=
1181:)
1178:0
1175:(
1168:f
1144:x
1140:a
1136:=
1133:)
1130:x
1127:(
1124:f
1100:x
1077:x
1073:e
1043:x
1020:x
1016:e
991:x
968:x
965:i
961:e
932:1
929:=
926:)
923:0
920:(
913:f
891:)
888:y
885:(
882:f
879:)
876:x
873:(
870:f
867:=
864:)
861:y
858:+
855:x
852:(
849:f
825:f
787:)
784:z
781:(
749:)
746:z
743:(
709:1
706:=
703:)
700:0
697:(
694:f
673:)
670:z
667:(
664:f
661:=
658:)
655:z
652:(
645:f
611:n
606:)
600:n
597:z
592:+
589:1
585:(
572:n
564:=
559:z
555:e
522:)
519:)
516:y
513:(
504:i
501:+
498:)
495:y
492:(
483:(
478:x
474:e
470:=
465:z
461:e
426:+
420:!
417:3
411:3
407:z
401:+
395:!
392:2
386:2
382:z
376:+
373:z
370:+
367:1
364:=
359:z
355:e
327:i
324:y
321:+
318:x
315:=
312:z
289:z
285:e
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