52:
676:
365:
2073:
of
Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this
2213:
2084:
2331: where all non-zero elements ≥1 and at least one of the elements of the set is 1. To find the period, T, first find the least common denominator of all the elements in the set. Period can be found as T =
2311:
643:
2251:
921:
1015:
1921:
458:
1506:
1720:
1416:
1195:
771:
724:
1369:
238:
2016:
1474:
2047:
1783:
1439:
1279:
947:
825:
794:
504:
402:
136:
1760:
1227:
1982:
1826:
1322:
76:
1944:
1803:
1740:
1667:
1629:
1599:
1579:
1559:
1342:
1299:
1247:
1139:
1119:
1099:
1079:
1059:
971:
666:
565:
481:
1848:, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form
340:. This definition of periodicity can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic
2208:{\displaystyle {\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}}
2395:
If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.
1604:
Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including:
1581:
that can be described by a
Fourier series, the coefficients of the series can be described by an integral over an interval of length
2645:
2268:
2321:
Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F =
2721:
2456:
2075:
2603:
2680:
1535:
2425:
831:
investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.
2731:
2675:
577:
2497:
2451:
45:
2726:
2518:
2222:
856:
2640:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) . Vol. 19. Berlin: Springer-Verlag. pp. x+247.
2472:
2430:
2405:
1541:. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If
321:
112:
979:
2477:
1854:
801:
410:
345:
325:
299:
94:
2670:
1523:
525:
is motion in which the position(s) of the system are expressible as periodic functions, all with the
279:.) Often, "the" period of a function is used to mean its fundamental period. A function with period
1479:
2066:
541:
1672:
1374:
1147:
729:
682:
51:
2550:
2546:
2467:
2446:
838:, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.
835:
193:
1841:
1347:
950:
1033:
are such functions. ("Incommensurate" in this context means not real multiples of each other.)
2716:
2689:
2641:
2542:
2487:
2462:
2435:
2254:
2058:
1987:
1444:
1030:
353:
38:
2021:
1829:
1531:
847:
98:
31:
2655:
1765:
1421:
1252:
929:
807:
776:
486:
384:
118:
2651:
2554:
2415:
2262:
1745:
1203:
568:
1961:
1808:
1304:
58:
2420:
2062:
1929:
1845:
1788:
1725:
1652:
1614:
1584:
1564:
1544:
1538:
1512:
1327:
1284:
1232:
1124:
1104:
1084:
1064:
1044:
1026:
956:
828:
651:
550:
466:
349:
2692:
2710:
669:
378:
298:
Geometrically, a periodic function can be defined as a function whose graph exhibits
141:, are periodic functions. Periodic functions are used throughout science to describe
2355:
For set representing all notes of
Western major scale: the LCD is 24 therefore T =
544:
can be formed from copies of one particular portion, repeated at regular intervals.
2633:
1041:
Periodic functions can take on values many times. More specifically, if a function
341:
2610:
2482:
2258:
2070:
1611:
taking a power or a root of a periodic function (provided it is defined for all
533:
142:
949:, the complex exponential is made up of cosine and sine waves. This means that
675:
2492:
2381:
For set representing all notes of a minor triad: the LCD is 10 therefore T =
2368:
For set representing all notes of a major triad: the LCD is 4 therefore T =
1608:
addition, subtraction, multiplication and division of periodic functions, and
2697:
2441:
2410:
1527:
150:
834:
According to the definition above, some exotic functions, for example the
1516:
102:
17:
2558:
537:
138:
926:
Since the cosine and sine functions are both periodic with period
674:
514:
363:
50:
44:"Aperiodic" and "Non-periodic" redirect here. For other uses, see
2578:
1029:
can have two incommensurate periods without being constant. The
518:
146:
27:
Function that repeats its values at regular intervals or periods
364:
2351:. Therefore, the LCD can be seen as a periodicity multiplier.
2306:{\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }
2065:
represent periodic functions and that
Fourier series satisfy
1958:
in this context. A periodic function is the special case
804:
sine and cosine are common periodic functions, with period
257:
of the function. If there exists a least positive constant
2604:"Periodicity, Real Fourier Series, and Fourier Transforms"
368:
A graph of the sine function, showing two complete periods
547:
A simple example of a periodic function is the function
291:, and these intervals are sometimes also referred to as
37:"Period length" redirects here. Not to be confused with
2519:"IEC 60050 — Details for IEV number 103-05-08: "cycle""
1482:
1350:
2271:
2225:
2087:
2024:
1990:
1964:
1932:
1857:
1811:
1791:
1768:
1748:
1728:
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1655:
1617:
1587:
1567:
1547:
1447:
1424:
1377:
1330:
1307:
1287:
1255:
1235:
1206:
1150:
1127:
1107:
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1067:
1047:
982:
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932:
859:
810:
779:
732:
685:
654:
580:
553:
489:
469:
413:
387:
196:
121:
61:
1984:, and an antiperiodic function is the special case
1840:A further generalization appears in the context of
571:" of its argument. Its period is 1. In particular,
55:An illustration of a periodic function with period
2305:
2245:
2207:
2041:
2010:
1976:
1938:
1915:
1820:
1797:
1777:
1754:
1734:
1714:
1661:
1623:
1593:
1573:
1553:
1500:
1468:
1433:
1410:
1363:
1336:
1316:
1293:
1273:
1241:
1221:
1189:
1133:
1113:
1093:
1073:
1053:
1009:
965:
941:
915:
819:
788:
765:
718:
660:
637:
559:
498:
475:
452:
396:
232:
130:
70:
2459:for computing periodicity in unevenly spaced data
1742:. For example, the sine and cosine functions are
1954:). Functions of this form are sometimes called
638:{\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}
509:Everyday examples are seen when the variable is
483:. This function repeats on intervals of length
348:can also be viewed as a function defined on the
97:that repeats its values at regular intervals or
2438:for computing periodicity in evenly spaced data
827:(see the figure on the right). The subject of
2557:), a least positive period may not exist (the
2313:is a representation of a 1-periodic function.
153:. Any function that is not periodic is called
2049:is rational, the function is also periodic.
8:
2202:
2185:
2145:
2142:
2136:
2108:
1645:One subset of periodic functions is that of
1511:Some periodic functions can be described by
2341:. Consider that for a simple sinusoid, T =
2246:{\displaystyle {\mathbb {R} /\mathbb {Z} }}
916:{\displaystyle e^{ikx}=\cos kx+i\,\sin kx.}
2638:Convexity methods in Hamiltonian mechanics
773:; both functions are periodic with period
2602:Summerson, Samantha R. (5 October 2009).
2523:International Electrotechnical Vocabulary
2299:
2298:
2290:
2289:
2284:
2280:
2279:
2278:
2270:
2238:
2237:
2232:
2228:
2227:
2226:
2224:
2198:
2197:
2181:
2180:
2161:
2160:
2132:
2131:
2118:
2117:
2100:
2099:
2094:
2090:
2089:
2088:
2086:
2031:
2023:
2000:
1989:
1963:
1931:
1883:
1856:
1810:
1790:
1767:
1747:
1727:
1674:
1654:
1616:
1586:
1566:
1546:
1483:
1481:
1446:
1423:
1376:
1351:
1349:
1329:
1306:
1286:
1254:
1234:
1205:
1149:
1126:
1106:
1086:
1066:
1046:
989:
981:
958:
931:
897:
864:
858:
809:
778:
731:
684:
653:
579:
552:
488:
468:
412:
386:
195:
120:
101:. The repeatable part of the function or
60:
2510:
356:these notions are defined accordingly.
253:for which this is the case is called a
953:(above) has the property such that if
1010:{\displaystyle L={\frac {2\pi }{k}}.}
263:with this property, it is called the
7:
1916:{\displaystyle f(x+P)=e^{ikP}f(x)~,}
1301:is a non-zero real number such that
973:is the period of the function, then
850:we have the common period function:
453:{\displaystyle \sin(x+2\pi )=\sin x}
1561:is a periodic function with period
285:will repeat on intervals of length
149:, and other phenomena that exhibit
249:in the domain. A nonzero constant
25:
1946:is a real or complex number (the
2074:end you can use the notion of a
2061:you encounter the problem, that
2457:Least-squares spectral analysis
1501:{\textstyle {\frac {2\pi }{5}}}
1025:A function whose domain is the
115:, which repeat at intervals of
2295:
1904:
1898:
1873:
1861:
1709:
1703:
1691:
1679:
1463:
1454:
1405:
1399:
1387:
1381:
1268:
1259:
1216:
1210:
1184:
1178:
1169:
1154:
760:
754:
742:
736:
713:
707:
695:
689:
620:
614:
605:
599:
590:
584:
513:; for instance the hands of a
506:(see the graph to the right).
435:
420:
227:
221:
212:
200:
1:
540:, that means that the entire
2426:Double Fourier sphere method
1805:-antiperiodic function is a
1715:{\displaystyle f(x+P)=-f(x)}
1536:almost everywhere convergent
1411:{\displaystyle f(x)=\sin(x)}
1190:{\displaystyle f(x+nP)=f(x)}
766:{\displaystyle g(x)=\cos(x)}
719:{\displaystyle f(x)=\sin(x)}
334:-direction by a distance of
2676:Encyclopedia of Mathematics
2541:For some functions, like a
1364:{\textstyle {\frac {P}{a}}}
233:{\displaystyle f(x+P)=f(x)}
2748:
2498:List of periodic functions
2452:Hill differential equation
1344:, is periodic with period
1229:is a function with period
1121:and all positive integers
648:The graph of the function
46:Aperiodic (disambiguation)
43:
36:
29:
2219:That is, each element in
2053:Quotient spaces as domain
1832:is not necessarily true.
1021:Double-periodic functions
521:show periodic behaviour.
2561:of all positive periods
2473:Periodic travelling wave
2431:Doubly periodic function
2406:Almost periodic function
2011:{\displaystyle k=\pi /P}
1836:Bloch-periodic functions
1828:-periodic function, the
1526:states that they have a
1469:{\displaystyle \sin(5x)}
1324:is within the domain of
1061:is periodic with period
381:is periodic with period
308:is periodic with period
30:Not to be confused with
2579:"Antiperiodic Function"
2265:. Thus a function like
2042:{\displaystyle kP/\pi }
842:Complex number examples
802:trigonometric functions
113:trigonometric functions
2722:Elementary mathematics
2478:Quasiperiodic function
2307:
2247:
2209:
2043:
2012:
1978:
1940:
1917:
1822:
1799:
1779:
1756:
1736:
1716:
1663:
1647:antiperiodic functions
1641:Antiperiodic functions
1625:
1595:
1575:
1555:
1502:
1470:
1435:
1412:
1365:
1338:
1318:
1295:
1275:
1243:
1223:
1191:
1135:
1115:
1095:
1075:
1055:
1011:
967:
943:
917:
821:
797:
790:
767:
720:
662:
639:
561:
532:For a function on the
500:
477:
454:
398:
369:
300:translational symmetry
234:
187:, it is the case that
132:
78:
72:
2583:mathworld.wolfram.com
2308:
2248:
2210:
2044:
2013:
1979:
1941:
1918:
1823:
1800:
1780:
1778:{\displaystyle 2\pi }
1757:
1737:
1717:
1664:
1649:. This is a function
1626:
1596:
1576:
1556:
1503:
1471:
1436:
1434:{\displaystyle 2\pi }
1413:
1366:
1339:
1319:
1296:
1276:
1274:{\displaystyle f(ax)}
1244:
1224:
1192:
1136:
1116:
1096:
1076:
1056:
1012:
968:
944:
942:{\displaystyle 2\pi }
918:
822:
820:{\displaystyle 2\pi }
791:
789:{\displaystyle 2\pi }
768:
721:
678:
663:
640:
562:
517:or the phases of the
501:
499:{\displaystyle 2\pi }
478:
455:
399:
397:{\displaystyle 2\pi }
367:
235:
133:
131:{\displaystyle 2\pi }
73:
54:
2269:
2261:that share the same
2223:
2085:
2067:convolution theorems
2022:
1988:
1962:
1930:
1855:
1809:
1789:
1766:
1755:{\displaystyle \pi }
1746:
1726:
1673:
1653:
1615:
1585:
1565:
1545:
1515:. For instance, for
1480:
1445:
1422:
1375:
1348:
1328:
1305:
1285:
1253:
1233:
1222:{\displaystyle f(x)}
1204:
1148:
1125:
1105:
1085:
1065:
1045:
980:
957:
930:
857:
808:
777:
730:
683:
652:
578:
551:
487:
467:
411:
385:
373:Real number examples
194:
119:
59:
2693:"Periodic Function"
2671:"Periodic function"
2577:Weisstein, Eric W.
1977:{\displaystyle k=0}
1785:-periodic. While a
111:. For example, the
2732:Types of functions
2690:Weisstein, Eric W.
2551:indicator function
2547:Dirichlet function
2468:Periodic summation
2447:Frequency spectrum
2317:Calculating period
2303:
2243:
2205:
2039:
2008:
1974:
1936:
1913:
1821:{\displaystyle 2P}
1818:
1795:
1775:
1762:-antiperiodic and
1752:
1732:
1712:
1659:
1621:
1591:
1571:
1551:
1524:Carleson's theorem
1498:
1466:
1431:
1408:
1361:
1334:
1317:{\displaystyle ax}
1314:
1291:
1271:
1239:
1219:
1187:
1131:
1111:
1091:
1071:
1051:
1031:elliptic functions
1007:
963:
939:
913:
836:Dirichlet function
817:
798:
786:
763:
716:
658:
635:
557:
496:
473:
463:for all values of
450:
394:
370:
302:, i.e. a function
265:fundamental period
243:for all values of
230:
128:
79:
71:{\displaystyle P.}
68:
2543:constant function
2488:Secular variation
2463:Periodic sequence
2436:Fourier transform
2255:equivalence class
2059:signal processing
1939:{\displaystyle k}
1909:
1798:{\displaystyle P}
1735:{\displaystyle x}
1662:{\displaystyle f}
1624:{\displaystyle x}
1594:{\displaystyle P}
1574:{\displaystyle P}
1554:{\displaystyle f}
1496:
1476:will have period
1359:
1337:{\displaystyle f}
1294:{\displaystyle a}
1242:{\displaystyle P}
1134:{\displaystyle n}
1114:{\displaystyle f}
1101:in the domain of
1094:{\displaystyle x}
1074:{\displaystyle P}
1054:{\displaystyle f}
1002:
966:{\displaystyle L}
848:complex variables
661:{\displaystyle f}
560:{\displaystyle f}
476:{\displaystyle x}
354:periodic sequence
295:of the function.
87:periodic waveform
83:periodic function
39:repeating decimal
16:(Redirected from
2739:
2727:Fourier analysis
2703:
2702:
2684:
2659:
2625:
2624:
2622:
2621:
2615:
2609:. Archived from
2608:
2599:
2593:
2592:
2590:
2589:
2574:
2568:
2566:
2555:rational numbers
2539:
2533:
2532:
2530:
2529:
2515:
2390:
2389:
2385:
2377:
2376:
2372:
2364:
2363:
2359:
2350:
2349:
2345:
2340:
2339:
2335:
2330:
2329:
2325:
2312:
2310:
2309:
2304:
2302:
2294:
2293:
2288:
2283:
2252:
2250:
2249:
2244:
2242:
2241:
2236:
2231:
2214:
2212:
2211:
2206:
2201:
2184:
2164:
2135:
2121:
2104:
2103:
2098:
2093:
2048:
2046:
2045:
2040:
2035:
2017:
2015:
2014:
2009:
2004:
1983:
1981:
1980:
1975:
1952:Floquet exponent
1948:Bloch wavevector
1945:
1943:
1942:
1937:
1922:
1920:
1919:
1914:
1907:
1894:
1893:
1842:Bloch's theorems
1827:
1825:
1824:
1819:
1804:
1802:
1801:
1796:
1784:
1782:
1781:
1776:
1761:
1759:
1758:
1753:
1741:
1739:
1738:
1733:
1721:
1719:
1718:
1713:
1668:
1666:
1665:
1660:
1630:
1628:
1627:
1622:
1600:
1598:
1597:
1592:
1580:
1578:
1577:
1572:
1560:
1558:
1557:
1552:
1507:
1505:
1504:
1499:
1497:
1492:
1484:
1475:
1473:
1472:
1467:
1441:and, therefore,
1440:
1438:
1437:
1432:
1417:
1415:
1414:
1409:
1370:
1368:
1367:
1362:
1360:
1352:
1343:
1341:
1340:
1335:
1323:
1321:
1320:
1315:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1248:
1246:
1245:
1240:
1228:
1226:
1225:
1220:
1196:
1194:
1193:
1188:
1140:
1138:
1137:
1132:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1016:
1014:
1013:
1008:
1003:
998:
990:
972:
970:
969:
964:
948:
946:
945:
940:
922:
920:
919:
914:
875:
874:
826:
824:
823:
818:
795:
793:
792:
787:
772:
770:
769:
764:
725:
723:
722:
717:
667:
665:
664:
659:
644:
642:
641:
636:
567:that gives the "
566:
564:
563:
558:
505:
503:
502:
497:
482:
480:
479:
474:
459:
457:
456:
451:
403:
401:
400:
395:
344:of the plane. A
339:
333:
319:
314:if the graph of
313:
307:
290:
284:
269:primitive period
262:
252:
248:
239:
237:
236:
231:
186:
172:
137:
135:
134:
129:
77:
75:
74:
69:
32:periodic mapping
21:
2747:
2746:
2742:
2741:
2740:
2738:
2737:
2736:
2707:
2706:
2688:
2687:
2669:
2666:
2648:
2636:(1990). "One".
2632:
2629:
2628:
2619:
2617:
2613:
2606:
2601:
2600:
2596:
2587:
2585:
2576:
2575:
2571:
2565:
2562:
2540:
2536:
2527:
2525:
2517:
2516:
2512:
2507:
2502:
2416:Continuous wave
2401:
2387:
2383:
2382:
2374:
2370:
2369:
2361:
2357:
2356:
2347:
2343:
2342:
2337:
2333:
2332:
2327:
2323:
2322:
2319:
2267:
2266:
2263:fractional part
2221:
2220:
2083:
2082:
2055:
2020:
2019:
1986:
1985:
1960:
1959:
1928:
1927:
1879:
1853:
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1807:
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1787:
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1764:
1763:
1744:
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1724:
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1671:
1670:
1651:
1650:
1643:
1638:
1636:Generalizations
1613:
1612:
1583:
1582:
1563:
1562:
1543:
1542:
1485:
1478:
1477:
1443:
1442:
1420:
1419:
1373:
1372:
1371:. For example,
1346:
1345:
1326:
1325:
1303:
1302:
1283:
1282:
1251:
1250:
1231:
1230:
1202:
1201:
1146:
1145:
1123:
1122:
1103:
1102:
1083:
1082:
1081:, then for all
1063:
1062:
1043:
1042:
1039:
1027:complex numbers
1023:
991:
978:
977:
955:
954:
951:Euler's formula
928:
927:
860:
855:
854:
844:
806:
805:
775:
774:
728:
727:
681:
680:
650:
649:
576:
575:
569:fractional part
549:
548:
523:Periodic motion
485:
484:
465:
464:
409:
408:
383:
382:
375:
362:
350:natural numbers
338:
335:
332:
329:
318:
315:
312:
309:
306:
303:
289:
286:
283:
280:
261:
258:
250:
247:
244:
192:
191:
185:
182:
171:
168:
165:
117:
116:
57:
56:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
2745:
2743:
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2734:
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2724:
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2709:
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2704:
2685:
2665:
2664:External links
2662:
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2423:
2421:Definite pitch
2418:
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2408:
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2400:
2397:
2393:
2392:
2379:
2366:
2318:
2315:
2301:
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2127:
2124:
2120:
2116:
2113:
2110:
2107:
2102:
2097:
2092:
2076:quotient space
2063:Fourier series
2054:
2051:
2038:
2034:
2030:
2027:
2007:
2003:
1999:
1996:
1993:
1973:
1970:
1967:
1956:Bloch-periodic
1935:
1924:
1923:
1912:
1906:
1903:
1900:
1897:
1892:
1889:
1886:
1882:
1878:
1875:
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1869:
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1863:
1860:
1846:Floquet theory
1837:
1834:
1817:
1814:
1794:
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1771:
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1637:
1634:
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1620:
1609:
1590:
1570:
1550:
1539:Fourier series
1513:Fourier series
1495:
1491:
1488:
1465:
1462:
1459:
1456:
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1450:
1430:
1427:
1407:
1404:
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829:Fourier series
816:
813:
785:
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735:
715:
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393:
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374:
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361:
358:
336:
330:
316:
310:
304:
287:
281:
259:
245:
241:
240:
229:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
183:
173:is said to be
169:
164:
161:
127:
124:
85:also called a
67:
64:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2744:
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2657:
2653:
2649:
2647:3-540-50613-6
2643:
2639:
2635:
2634:Ekeland, Ivar
2631:
2630:
2616:on 2019-08-25
2612:
2605:
2598:
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1128:
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992:
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849:
841:
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832:
830:
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783:
780:
757:
751:
748:
745:
739:
733:
710:
704:
701:
698:
692:
686:
677:
673:
671:
670:sawtooth wave
655:
632:
629:
626:
623:
617:
611:
608:
602:
596:
593:
587:
581:
574:
573:
572:
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520:
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493:
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447:
444:
441:
438:
432:
429:
426:
423:
417:
414:
407:
406:
405:
391:
388:
380:
379:sine function
372:
366:
359:
357:
355:
351:
347:
343:
342:tessellations
327:
323:
301:
296:
294:
278:
274:
270:
266:
256:
224:
218:
215:
209:
206:
203:
197:
190:
189:
188:
180:
177:if, for some
176:
162:
160:
158:
157:
152:
148:
144:
140:
125:
122:
114:
110:
109:
104:
100:
96:
92:
91:periodic wave
88:
84:
65:
62:
53:
47:
40:
33:
19:
2696:
2674:
2637:
2618:. Retrieved
2611:the original
2597:
2586:. Retrieved
2582:
2572:
2567:being zero).
2537:
2526:. Retrieved
2522:
2513:
2394:
2320:
2259:real numbers
2218:
2056:
1955:
1951:
1947:
1925:
1839:
1646:
1644:
1603:
1517:
1510:
1199:
1040:
1024:
925:
845:
833:
799:
647:
546:
534:real numbers
531:
526:
522:
510:
508:
462:
376:
352:, and for a
297:
292:
277:prime period
276:
273:basic period
272:
268:
264:
254:
242:
178:
174:
166:
155:
154:
143:oscillations
107:
106:
105:is called a
90:
86:
82:
80:
2483:Seasonality
2071:convolution
2018:. Whenever
1418:has period
326:translation
167:A function
151:periodicity
89:(or simply
2711:Categories
2620:2018-03-24
2588:2024-06-06
2528:2023-11-20
2505:References
2493:Wavelength
1669:such that
1037:Properties
679:A plot of
536:or on the
163:Definition
2698:MathWorld
2681:EMS Press
2442:Frequency
2411:Amplitude
2296:→
2195:∈
2178:∈
2172:−
2166:∧
2158:∈
2129:∈
2037:π
1998:π
1773:π
1750:π
1698:−
1528:pointwise
1520:functions
1490:π
1452:
1429:π
1397:
996:π
937:π
902:
883:
815:π
784:π
752:
705:
627:⋯
494:π
445:
433:π
418:
392:π
322:invariant
181:constant
156:aperiodic
126:π
18:Aperiodic
2717:Calculus
2683:. 2001 .
2399:See also
1830:converse
1722:for all
1532:Lebesgue
1281:, where
538:integers
529:period.
404:, since
360:Examples
346:sequence
175:periodic
103:waveform
95:function
93:), is a
2656:1051888
2559:infimum
2553:of the
2545:or the
2386:⁄
2373:⁄
2360:⁄
2346:⁄
2336:⁄
2326:⁄
1249:, then
668:is the
328:in the
293:periods
179:nonzero
139:radians
99:periods
2654:
2644:
2253:is an
2069:(i.e.
1926:where
1908:
846:Using
324:under
267:(also
255:period
2614:(PDF)
2607:(PDF)
2549:(the
542:graph
515:clock
275:, or
147:waves
108:cycle
2642:ISBN
1844:and
800:The
726:and
527:same
519:moon
511:time
377:The
2334:LCD
2257:of
2057:In
1950:or
1449:sin
1394:sin
1200:If
899:sin
880:cos
749:cos
702:sin
633:0.5
618:2.5
603:1.5
588:0.5
442:sin
415:sin
320:is
2713::
2695:.
2679:.
2673:.
2652:MR
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48:.
41:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
