31:
1537:
5229:
5030:
4638:
2849:
5025:{\displaystyle {\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)\left(\cos(2\pi \xi t)-i\,\sin(2\pi \xi t)\right)dt&&{\text{Euler's Formula}}\\&=\left(\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt\right)-i\left(\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt\right)\\&={\hat {f}}^{c}(\xi )-i\,{\hat {f}}^{s}(\xi )\,.\end{aligned}}}
119:
813:
1338:
452:
4226:
1899:
2600:
5293:
of the oscillation are required, an example of which is Ooura's method for
Fourier integrals This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.
5292:
Using standard methods of numerical evaluation for
Fourier integrals, such as Gaussian or tanh-sinh quadrature, is likely to lead to completely incorrect results, as the quadrature sum is (for most integrands of interest) highly ill-conditioned. Special numerical methods which exploit the structure
5263:
needed in the regular
Fourier transform can be avoided. They may also be convenient when the original function is already even or odd or can be made even or odd, in which case only the cosine or the sine transform respectively is needed. For instance, even though an input may not be even or odd, a
3924:
4634:
it can be shown (for real-valued functions) that the
Fourier transform's real component is the cosine transform (representing the even component of the original function) and the Fourier transform's imaginary component is the negative of the sine transform (representing the odd component of the
3073:
1134:
1698:
2044:
1532:
3994:
3395:
3233:
4539:
2844:{\displaystyle f(t)=\underbrace {\int _{-\infty }^{\infty }{\hat {f}}^{s}(\xi )\sin(2\pi \xi t)\,d\xi } _{{\text{odd component of }}f(t)}\,+\underbrace {\int _{-\infty }^{\infty }{\hat {f}}^{c}(\xi )\cos(2\pi \xi t)\,d\xi } _{{\text{even component of }}f(t)}\,.}
3682:
960:
4412:
2301:
2180:
122:
Fourier transforms relate a time-domain function (red) to a frequency-domain function (blue). Sine or cosine waves that make up the original function will appear as peaks in the frequency domain functions produced by the sine or cosine transform,
3731:
2884:
1908:
5710:
1396:
2503:
2402:
692:
274:
3266:
3104:
4428:
1333:{\displaystyle {\hat {f}}^{c}(\xi )=\int _{-\infty }^{\infty }\overbrace {f_{\text{even}}(t)\cdot \cos(2\pi \xi t)} ^{\text{even·even=even}}\,dt=2\int _{0}^{\infty }f_{\text{even}}(t)\cos(2\pi \xi t)\,dt.}
447:
803:
4221:{\displaystyle 2\int _{-\infty }^{\infty }\int _{0}^{\infty }e^{-\delta \xi }\cos(2\pi \xi (x-t))\,d\xi \,f(x)\,dx=\int _{-\infty }^{\infty }f(x){\frac {2\delta }{\delta ^{2}+4\pi ^{2}(x-t)^{2}}}\,dx.}
1894:{\displaystyle {\hat {f}}^{s}(\xi )=\int _{-\infty }^{\infty }\overbrace {f_{\text{odd}}(t)\cdot \sin(2\pi \xi t)} ^{\text{odd·odd=even}}\,dt=2\int _{0}^{\infty }f_{\text{odd}}(t)\sin(2\pi \xi t)\,dt}
5219:
5121:
3554:
864:
4643:
4433:
4283:
2185:
2067:
5551:
5369:
4632:
1019:
1671:
858:
3963:
2407:
2306:
5256:
amplitude information inside its complex valued result. But a disadvantage is its requirement on understanding complex numbers, complex exponentials, and negative frequency.
486:
5888:
A Course Of Modern
Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions
5587:
5405:
3529:
3493:
4252:
3989:
1592:
1627:
1365:
1044:
5502:
5259:
The sine and cosine transforms meanwhile have the advantage that all quantities are real. Since positive frequencies can fully express them, the non-trivial concept of
5470:
2523:
1385:
1107:
1064:
538:
5930:
5638:
5434:
3437:
2877:
719:
576:
360:
321:
158:
5830:
4278:
5744:
5616:
4559:
3726:
3702:
3457:
3417:
3255:
3095:
2591:
2563:
2543:
1691:
1127:
1087:
506:
297:
3531:). A consequence of this symmetry is that their inversion and transform processes still work when the two functions are swapped. Two such functions are called
585:
167:
3919:{\displaystyle {\tfrac {1}{2}}\lim _{h\to 0}\left(f(t+h)+f(t-h)\right)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(x)\cos(2\pi \xi (x-t))\,dx\,d\xi .}
3068:{\displaystyle f(t)=2\int _{0}^{\infty }{\hat {f}}^{s}(\xi )\sin(2\pi \xi t)\,d\xi \,+2\int _{0}^{\infty }{\hat {f}}^{c}(\xi )\cos(2\pi \xi t)\,d\xi \,.}
3928:
This latter form is a useful intermediate step in proving the inverse formulae for the since and cosine transforms. One method of deriving it, due to
2039:{\displaystyle {\hat {f}}^{s}(\xi )=\int _{-\infty }^{\infty }\overbrace {f_{\text{even}}(t)\cdot \sin(2\pi \xi t)} ^{\text{even·odd=odd}}\,dt=0.}
367:
1527:{\displaystyle {\hat {f}}^{c}(\xi )=\int _{-\infty }^{\infty }\overbrace {f_{\text{odd}}(t)\cdot \cos(2\pi \xi t)} ^{\text{odd·even=odd}}\,dt=0.}
726:
1536:
5130:
3399:
Remarkably, these last two simplified inversion formulas look identical to the original sine and cosine transforms, respectively, though with
5905:
5806:
5039:
30:
5312:
5895:
5753:
5766:
3390:{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}^{c}(\xi )\cos(2\pi \xi t)\,d\xi ,{\text{ only if }}f(t){\text{ is even.}}}
3228:{\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}^{s}(\xi )\sin(2\pi \xi t)\,d\xi ,{\text{ only if }}f(t){\text{ is odd.}}}
5963:
5237:
5248:
An advantage of the modern
Fourier transform is that while the sine and cosine transforms together are required to extract the
118:
812:
6015:
5920:
4534:{\displaystyle {\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\xi t}\,dt\\\end{aligned}}\,}
6005:
2879:, the concept of negative frequency can be avoided by doubling the result of integrating over non-negative frequencies:
5635:
has this symmetry even when the original functions aren't even or odd. A notation to denote
Fourier transform pairs is
4562:
5281:
6010:
5619:
5302:
5265:
509:
82:
both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of
5618:
and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see
5228:
3677:{\displaystyle f(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x)\cos(2\pi \xi (x-t))\,dx\,d\xi .}
2593:
can be recovered from its sine and cosine transforms under the usual hypotheses using the inversion formula:
955:{\displaystyle {\hat {f}}^{c}(\xi )={\sqrt {\tfrac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}
5879:
5307:
5277:
5273:
5269:
5238:
same frequency, but whose amplitude and phase depends on the amplitudes of the original sine and cosine wave
5516:
5334:
2060:
Just like the
Fourier transform takes the form of different equations with different constant factors (see
5032:
Because of this relationship, the cosine transform of functions whose
Fourier transform is known (e.g. in
4571:
4407:{\displaystyle f(t)\int _{-\infty }^{\infty }{\frac {2\delta }{\delta ^{2}+4\pi ^{2}(x-t)^{2}}}\,dx=f(t).}
2296:{\displaystyle {\hat {f}}^{s}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\sin(2\pi \xi t)\,dt.}
2049:
1541:
1388:
1066:
982:
981:
shown in the overbraces in the following equations dramatically simplify the integrands when transforming
978:
71:
2175:{\displaystyle {\hat {f}}^{c}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\cos(2\pi \xi t)\,dt}
5883:
988:
1640:
823:
3935:
1553:
1549:
462:
456:
5556:
5374:
3498:
3462:
328:
4231:
3968:
1559:
820:
is a mirror image of its right half and its sine transform is entirely 0. Gaussians have the form
5824:
5475:
5260:
1629:
is the above plot. Thus, the sine wave function and the time-shifted Dirac delta function form a
1597:
91:
63:
4566:
1345:
1024:
83:
5481:
5901:
5891:
5844:
5812:
5802:
5749:
5632:
5445:
5033:
4422:
3705:
2565:
is often instead used to represent a spatial domain when transforming to spatial frequencies.
2061:
817:
332:
99:
75:
74:
of the function plus cosine waves representing the even component of the function. The modern
5705:{\displaystyle f(t)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ).}
5455:
2508:
1370:
1092:
1049:
514:
5848:
5252:
information of a frequency, the modern
Fourier transform instead compactly packs both phase
107:
35:
327:
in cycles per unit time, but in the abstract, they can be any dual pair of variables (e.g.
5419:
3422:
2862:
1552:. Their cosine transform is entirely zero. The above odd function contains two half-sized
704:
552:
345:
306:
134:
5774:
4257:
2498:{\displaystyle F_{s}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\sin(\alpha x)\,dx}
2397:{\displaystyle F_{c}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\cos(\alpha x)\,dx}
687:{\displaystyle {\hat {f}}^{c}(\xi )=\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt.}
269:{\displaystyle {\hat {f}}^{s}(\xi )=\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt.}
5601:
4544:
3711:
3687:
3442:
3402:
3240:
3080:
2576:
2548:
2528:
1676:
1545:
1112:
1072:
491:
282:
95:
87:
5952:
5999:
5249:
5233:
3258:
1021:. Since cosine is an even function and because the integral of an even function from
698:
5739:
3098:
339:
5232:
Adding a sine wave (red) and a cosine wave (blue) of the same frequency results a
2062:
Fourier transform § Unitarity and definition for square integrable functions
5240:. Hence, at a particular frequency, the sine transform and the cosine transform
1109:, the cosine transform of any even function can be simplified to avoid negative
55:
17:
5925:
103:
67:
5816:
324:
39:
5990:, Journal of computational and applied mathematics 112.1-2 (1999): 229-241.
5796:
5244:
essentially only represent one sine wave that could have any phase shift.
5890:(4th ed.). Cambridge, UK: Cambridge University Press. p. 189.
3261:, then the sine transform is zero, so its inversion also simplifies to:
985:. Some authors even only define the cosine transform for even functions
5449:
5036:) can be simply found by taking the real part of the Fourier transform:
1637:
Similarly, because sin is odd, the sine transform of any odd function
5441:
3929:
3544:
2052:, while the cosine transform represents the even part of a function.
3101:, then the cosine transform is zero, so its inversion simplifies to:
2064:
for discussion), other authors also define the cosine transform as
451:
5437:
5227:
1535:
811:
450:
117:
47:
29:
300:
98:'s original transform equations and are still preferred in some
43:
5034:
Fourier transform § Tables of important Fourier transforms
5988:
A robust double exponential formula for Fourier-type integrals
3684:
This theorem is often stated under different hypotheses, that
5798:
An Introduction to Partial Differential Equations with MATLAB
5667:
5523:
5341:
1903:
and the sine transform of any even function is simply zero:
442:{\displaystyle {\hat {f}}^{s}(-\xi )=-{\hat {f}}^{s}(\xi ).}
106:
applications and may be better suited as an introduction to
5728:, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211
1391:, the cosine transform of any odd function is simply zero:
798:{\displaystyle {\hat {f}}^{c}(\xi )={\hat {f}}^{c}(-\xi ).}
5444:
as units, these transforms are sometimes expressed using
34:
The sine and cosine transforms convert a function into a
5214:{\displaystyle {\hat {f}}^{s}(\xi )=-\mathrm {Im} {}\,.}
2596:
Fourier inversion (from the sine and cosine transforms)
79:
4574:
3736:
3547:, the full inversion formula can also be rewritten as
2859:
Note that since both integrands are even functions of
901:
467:
5641:
5604:
5559:
5519:
5484:
5458:
5422:
5377:
5337:
5133:
5042:
4641:
4547:
4431:
4286:
4260:
4234:
3997:
3971:
3938:
3734:
3714:
3690:
3557:
3501:
3465:
3445:
3425:
3405:
3269:
3243:
3107:
3083:
2887:
2865:
2603:
2579:
2551:
2531:
2511:
2410:
2309:
2188:
2070:
1911:
1701:
1679:
1643:
1600:
1562:
1399:
1373:
1348:
1137:
1115:
1095:
1075:
1052:
1027:
991:
867:
826:
729:
707:
588:
555:
517:
494:
465:
370:
348:
309:
285:
170:
137:
5921:"Highlights in the History of the Fourier Transform"
5280:
of its input, to avoid having to compute the entire
5116:{\displaystyle {\hat {f}}^{c}(\xi )=\mathrm {Re} {}}
2303:Another convention defines the cosine transform as
5866:Introduction to the theory of the Fourier integral
5850:Theorie analytique de la propagation de la chaleur
5704:
5610:
5581:
5545:
5496:
5464:
5436:in cycles per unit time, which typically uses the
5428:
5399:
5363:
5213:
5115:
5024:
4626:
4553:
4533:
4406:
4272:
4246:
4220:
3983:
3957:
3918:
3720:
3696:
3676:
3523:
3487:
3451:
3431:
3411:
3389:
3249:
3227:
3089:
3067:
2871:
2843:
2585:
2557:
2537:
2517:
2497:
2396:
2295:
2174:
2038:
1893:
1685:
1665:
1621:
1586:
1526:
1379:
1359:
1332:
1121:
1101:
1081:
1058:
1038:
1013:
954:
852:
797:
713:
686:
570:
532:
500:
480:
441:
354:
315:
291:
268:
152:
66:that decompose arbitrary functions into a sum of
3748:
2545:is typically used to represent the time domain,
5767:"Fourier Transform, Cosine and Sine Transforms"
5513:The cosine transform is sometimes denoted with
5416:While this article uses ordinary frequency for
5127:of the imaginary part of the Fourier transform:
979:multiplication rules for even and odd functions
963:also is a Gaussian. The plotted Gaussian uses
5745:Mathematical Methods in the Physical Sciences
5331:The sine transform is sometimes denoted with
8:
5801:(Second ed.). Boca Raton. p. 221.
816:Like all even functions, the left half of a
5748:, 2nd Ed, John Wiley & Sons Inc, 1983.
42:. The inverse transform converts back to a
5868:, Oxford at the Clarendon Press, p. 1
5829:: CS1 maint: location missing publisher (
5202:
5180:
5108:
5086:
2525:as the transformation variable. And while
5679:
5678:
5666:
5665:
5660:
5658:
5657:
5640:
5603:
5573:
5562:
5561:
5558:
5528:
5522:
5521:
5518:
5483:
5457:
5421:
5391:
5380:
5379:
5376:
5346:
5340:
5339:
5336:
5207:
5182:
5181:
5176:
5168:
5147:
5136:
5135:
5132:
5088:
5087:
5082:
5074:
5056:
5045:
5044:
5041:
5014:
4999:
4988:
4987:
4985:
4964:
4953:
4952:
4929:
4887:
4879:
4852:
4810:
4802:
4778:
4739:
4686:
4678:
4647:
4646:
4642:
4640:
4582:
4573:
4546:
4530:
4519:
4498:
4476:
4468:
4437:
4436:
4432:
4430:
4379:
4370:
4348:
4332:
4317:
4311:
4303:
4285:
4259:
4233:
4208:
4199:
4177:
4161:
4146:
4128:
4120:
4106:
4093:
4086:
4038:
4028:
4023:
4013:
4005:
3996:
3970:
3943:
3937:
3906:
3899:
3845:
3837:
3827:
3822:
3751:
3735:
3733:
3713:
3708:on an open interval containing the point
3689:
3664:
3657:
3603:
3595:
3585:
3577:
3556:
3515:
3504:
3503:
3500:
3479:
3468:
3467:
3464:
3444:
3424:
3404:
3382:
3365:
3355:
3316:
3305:
3304:
3297:
3289:
3268:
3242:
3220:
3203:
3193:
3154:
3143:
3142:
3135:
3127:
3106:
3082:
3061:
3054:
3015:
3004:
3003:
2996:
2991:
2980:
2973:
2934:
2923:
2922:
2915:
2910:
2886:
2864:
2837:
2818:
2817:
2804:
2765:
2754:
2753:
2746:
2738:
2731:
2726:
2707:
2706:
2693:
2654:
2643:
2642:
2635:
2627:
2620:
2602:
2578:
2550:
2530:
2510:
2488:
2452:
2447:
2433:
2415:
2409:
2387:
2351:
2346:
2332:
2314:
2308:
2283:
2241:
2236:
2220:
2202:
2191:
2190:
2187:
2165:
2123:
2118:
2102:
2084:
2073:
2072:
2069:
2023:
2017:
1969:
1962:
1955:
1947:
1925:
1914:
1913:
1910:
1884:
1845:
1835:
1830:
1813:
1807:
1759:
1752:
1745:
1737:
1715:
1704:
1703:
1700:
1678:
1648:
1642:
1599:
1561:
1511:
1505:
1457:
1450:
1443:
1435:
1413:
1402:
1401:
1398:
1372:
1349:
1347:
1320:
1281:
1271:
1266:
1249:
1243:
1195:
1188:
1181:
1173:
1151:
1140:
1139:
1136:
1114:
1094:
1074:
1051:
1028:
1026:
996:
990:
938:
922:
918:
913:
899:
881:
870:
869:
866:
842:
831:
825:
774:
763:
762:
743:
732:
731:
728:
706:
674:
632:
624:
602:
591:
590:
587:
554:
516:
493:
466:
464:
421:
410:
409:
384:
373:
372:
369:
347:
308:
284:
256:
214:
206:
184:
173:
172:
169:
136:
4254:, the integrand tends to zero except at
5732:
5324:
5123:while the sine transform is simply the
697:The cosine transform is necessarily an
5822:
5724:Whittaker, Edmund, and James Watson,
5546:{\displaystyle {\mathcal {F}}_{c}(f)}
5364:{\displaystyle {\mathcal {F}}_{s}(f)}
4627:{\textstyle (e^{ix}=\cos x+i\sin x),}
338:The sine transform is necessarily an
7:
4421:The complex exponential form of the
5919:Valentinuzzi, Max E. (2016-01-25).
3237:Likewise, if the original function
5853:. Paris: G. Carré. pp. 108ff.
5313:List of Fourier-related transforms
5172:
5169:
5078:
5075:
4888:
4883:
4811:
4806:
4687:
4682:
4477:
4472:
4417:Relation with complex exponentials
4312:
4307:
4129:
4124:
4029:
4014:
4009:
3846:
3841:
3828:
3604:
3599:
3586:
3581:
3298:
3293:
3136:
3131:
2997:
2916:
2747:
2742:
2636:
2631:
2453:
2352:
2242:
2124:
2048:The sine transform represents the
1956:
1951:
1836:
1746:
1741:
1673:also simplifies to avoid negative
1554:time-shifted Dirac delta functions
1444:
1439:
1374:
1354:
1272:
1182:
1177:
1096:
1053:
1033:
1014:{\displaystyle f_{\text{even}}(t)}
633:
628:
215:
210:
94:, they more closely correspond to
25:
5951:Williams, Lance R. (2011-09-06).
1666:{\displaystyle f_{\text{odd}}(t)}
853:{\displaystyle e^{-\alpha t^{2}}}
455:The cosine transform of a simple
4280:, so that formally the above is
3958:{\displaystyle e^{-\delta \xi }}
1594:Likewise, the sine transform of
973:and is its own cosine transform.
5969:from the original on 2024-05-02
5933:from the original on 2024-05-15
3543:Using the addition formula for
1556:. Its sine transform is simply
1540:Odd functions are unchanged if
481:{\displaystyle {\tfrac {1}{a}}}
5696:
5690:
5661:
5651:
5645:
5598:The usual hypotheses are that
5582:{\displaystyle {\hat {f}}^{c}}
5567:
5540:
5534:
5400:{\displaystyle {\hat {f}}^{s}}
5385:
5358:
5352:
5203:
5199:
5193:
5187:
5177:
5159:
5153:
5141:
5109:
5105:
5099:
5093:
5083:
5068:
5062:
5050:
5011:
5005:
4993:
4976:
4970:
4958:
4926:
4911:
4902:
4896:
4849:
4834:
4825:
4819:
4761:
4746:
4730:
4715:
4701:
4695:
4664:
4658:
4652:
4618:
4575:
4491:
4485:
4454:
4448:
4442:
4398:
4392:
4367:
4354:
4296:
4290:
4238:
4196:
4183:
4143:
4137:
4103:
4097:
4083:
4080:
4068:
4056:
3896:
3893:
3881:
3869:
3860:
3854:
3804:
3792:
3783:
3771:
3755:
3654:
3651:
3639:
3627:
3618:
3612:
3567:
3561:
3524:{\displaystyle {\hat {f}}^{c}}
3509:
3488:{\displaystyle {\hat {f}}^{s}}
3473:
3379:
3373:
3352:
3337:
3328:
3322:
3310:
3279:
3273:
3217:
3211:
3190:
3175:
3166:
3160:
3148:
3117:
3111:
3051:
3036:
3027:
3021:
3009:
2970:
2955:
2946:
2940:
2928:
2897:
2891:
2832:
2826:
2801:
2786:
2777:
2771:
2759:
2721:
2715:
2690:
2675:
2666:
2660:
2648:
2613:
2607:
2485:
2476:
2467:
2461:
2427:
2421:
2384:
2375:
2366:
2360:
2326:
2320:
2280:
2265:
2256:
2250:
2214:
2208:
2196:
2162:
2147:
2138:
2132:
2096:
2090:
2078:
2008:
1993:
1981:
1975:
1937:
1931:
1919:
1881:
1866:
1857:
1851:
1798:
1783:
1771:
1765:
1727:
1721:
1709:
1660:
1654:
1616:
1607:
1578:
1569:
1496:
1481:
1469:
1463:
1425:
1419:
1407:
1342:And because the integral from
1317:
1302:
1293:
1287:
1234:
1219:
1207:
1201:
1163:
1157:
1145:
1008:
1002:
935:
925:
893:
887:
875:
789:
780:
768:
755:
749:
737:
671:
656:
647:
641:
614:
608:
596:
565:
559:
527:
518:
433:
427:
415:
399:
390:
378:
253:
238:
229:
223:
196:
190:
178:
147:
141:
1:
5986:Takuya Ooura, Masatake Mori,
1389:any odd function from is zero
5795:Coleman, Matthew P. (2013).
4247:{\displaystyle \delta \to 0}
3984:{\displaystyle \delta >0}
1587:{\displaystyle \sin(a\xi ).}
860:and their cosine transform:
5726:A Course in Modern Analysis
4563:square root of negative one
3539:Overview of inversion proof
1622:{\displaystyle \sin(a\xi )}
808:Odd and even simplification
701:of frequency, i.e. for all
342:of frequency, i.e. for all
38:representation as a sum of
6032:
5282:discrete Fourier transform
5270:assuming an even extension
5236:sine wave (green) of that
3549:Fourier's integral formula
2404:and the sine transform as
2182:and the sine transform as
1360:{\displaystyle {-}\infty }
1039:{\displaystyle {-}\infty }
60:sine and cosine transforms
27:Variant Fourier transforms
5960:www.cs.unm.edu/~williams/
5864:Edwin Titchmarsh (1948),
5620:Fourier inversion theorem
5497:{\displaystyle 2\pi \xi }
5303:Discrete cosine transform
5278:assuming an odd extension
5266:discrete cosine transform
4425:used more often today is
3965:into the integral, where
3704:is integrable, and is of
581:Fourier cosine transform
5953:"Even and odd functions"
5880:Whittaker, Edmund Taylor
5765:Nyack, Cuthbert (1996).
5631:The more general modern
547:Fourier cosine transform
5465:{\displaystyle \omega }
5452:) per unit time, where
5448:in angular units (e.g.
5308:Discrete sine transform
5274:discrete sine transform
2820:even component of
2518:{\displaystyle \alpha }
1380:{\displaystyle \infty }
1102:{\displaystyle \infty }
1059:{\displaystyle \infty }
533:{\displaystyle (a\xi )}
163:Fourier sine transform
5884:Watson, George Neville
5706:
5612:
5583:
5547:
5498:
5466:
5430:
5401:
5365:
5245:
5215:
5117:
5026:
4628:
4555:
4535:
4408:
4274:
4248:
4222:
3985:
3959:
3920:
3722:
3698:
3678:
3525:
3489:
3453:
3433:
3413:
3391:
3251:
3229:
3091:
3069:
2873:
2845:
2709:odd component of
2587:
2573:The original function
2559:
2539:
2519:
2499:
2398:
2297:
2176:
2050:odd part of a function
2040:
1895:
1687:
1667:
1634:
1623:
1588:
1528:
1381:
1361:
1334:
1123:
1103:
1083:
1060:
1040:
1015:
983:even and odd functions
974:
956:
854:
799:
715:
688:
572:
541:
534:
502:
482:
443:
356:
317:
293:
270:
154:
129:Fourier sine transform
124:
51:
5707:
5613:
5584:
5548:
5499:
5467:
5431:
5402:
5366:
5272:of its input while a
5231:
5216:
5118:
5027:
4629:
4556:
4536:
4409:
4275:
4249:
4223:
3986:
3960:
3921:
3723:
3699:
3679:
3526:
3490:
3454:
3434:
3414:
3392:
3252:
3230:
3092:
3070:
2874:
2846:
2588:
2560:
2540:
2520:
2500:
2399:
2298:
2177:
2041:
1896:
1688:
1668:
1624:
1589:
1539:
1529:
1382:
1362:
1335:
1124:
1104:
1084:
1067:is twice its integral
1061:
1041:
1016:
957:
855:
815:
800:
716:
689:
573:
535:
503:
483:
454:
444:
357:
318:
294:
271:
155:
121:
40:sine and cosine waves
33:
6016:Mathematical physics
5771:cnyack.homestead.com
5639:
5602:
5557:
5517:
5482:
5456:
5429:{\displaystyle \xi }
5420:
5375:
5335:
5288:Numerical evaluation
5131:
5040:
4639:
4572:
4545:
4429:
4284:
4258:
4232:
3995:
3969:
3936:
3732:
3712:
3688:
3555:
3499:
3463:
3443:
3432:{\displaystyle \xi }
3423:
3403:
3267:
3241:
3105:
3081:
2885:
2872:{\displaystyle \xi }
2863:
2601:
2577:
2549:
2529:
2509:
2408:
2307:
2186:
2068:
1909:
1699:
1677:
1641:
1598:
1560:
1397:
1371:
1346:
1135:
1113:
1093:
1073:
1050:
1025:
989:
865:
824:
727:
714:{\displaystyle \xi }
705:
586:
571:{\displaystyle f(t)}
553:
515:
492:
463:
457:rectangular function
368:
355:{\displaystyle \xi }
346:
316:{\displaystyle \xi }
307:
283:
168:
153:{\displaystyle f(t)}
135:
84:complex exponentials
6006:Integral transforms
4892:
4815:
4691:
4635:original function):
4481:
4316:
4273:{\displaystyle x=t}
4133:
4033:
4018:
3850:
3832:
3608:
3590:
3367: only if
3302:
3205: only if
3140:
3001:
2920:
2751:
2640:
2457:
2356:
2246:
2128:
1960:
1840:
1750:
1448:
1276:
1245:even·even=even
1186:
637:
219:
5702:
5608:
5579:
5543:
5494:
5476:radians per second
5462:
5426:
5397:
5361:
5261:negative frequency
5246:
5211:
5113:
5022:
5020:
4875:
4798:
4674:
4624:
4551:
4531:
4528:
4464:
4404:
4299:
4270:
4244:
4218:
4116:
4019:
4001:
3981:
3955:
3916:
3833:
3818:
3762:
3745:
3718:
3694:
3674:
3591:
3573:
3521:
3485:
3449:
3429:
3409:
3387:
3285:
3247:
3225:
3123:
3087:
3065:
2987:
2906:
2869:
2841:
2836:
2815:
2734:
2725:
2704:
2623:
2583:
2555:
2535:
2515:
2495:
2443:
2394:
2342:
2293:
2232:
2172:
2114:
2036:
1943:
1891:
1826:
1733:
1683:
1663:
1635:
1619:
1584:
1524:
1431:
1377:
1357:
1330:
1262:
1169:
1119:
1099:
1079:
1056:
1036:
1011:
975:
952:
910:
850:
795:
711:
684:
620:
568:
542:
530:
498:
478:
476:
439:
352:
313:
289:
266:
202:
150:
125:
92:negative frequency
86:and don't require
64:integral equations
52:
5906:978-0-521-06794-2
5808:978-1-4398-9846-8
5687:
5677:
5672:
5656:
5633:Fourier transform
5611:{\displaystyle f}
5570:
5446:angular frequency
5388:
5190:
5144:
5096:
5053:
4996:
4961:
4781:
4655:
4554:{\displaystyle i}
4445:
4423:Fourier transform
4377:
4206:
3747:
3744:
3721:{\displaystyle t}
3706:bounded variation
3697:{\displaystyle f}
3512:
3476:
3452:{\displaystyle f}
3412:{\displaystyle t}
3385:
3368:
3313:
3250:{\displaystyle f}
3223:
3206:
3151:
3090:{\displaystyle f}
3012:
2931:
2821:
2762:
2732:
2730:
2710:
2651:
2621:
2619:
2586:{\displaystyle f}
2569:Fourier inversion
2558:{\displaystyle x}
2538:{\displaystyle t}
2441:
2340:
2230:
2229:
2199:
2112:
2111:
2081:
2056:Other conventions
2022:
2020:
2019:even·odd=odd
2015:
1972:
1922:
1848:
1812:
1810:
1809:odd·odd=even
1805:
1762:
1712:
1686:{\displaystyle t}
1651:
1510:
1508:
1507:odd·even=odd
1503:
1460:
1410:
1284:
1248:
1246:
1241:
1198:
1148:
1122:{\displaystyle t}
1082:{\displaystyle 0}
999:
948:
911:
909:
878:
818:Gaussian function
771:
740:
599:
501:{\displaystyle a}
475:
418:
381:
333:spatial frequency
292:{\displaystyle t}
181:
100:signal processing
76:Fourier transform
70:representing the
16:(Redirected from
6023:
6011:Fourier analysis
5991:
5984:
5978:
5977:
5975:
5974:
5968:
5957:
5948:
5942:
5941:
5939:
5938:
5916:
5910:
5909:
5876:
5870:
5869:
5861:
5855:
5854:
5841:
5835:
5834:
5828:
5820:
5792:
5786:
5785:
5783:
5782:
5773:. Archived from
5762:
5756:
5737:
5712:
5711:
5709:
5708:
5703:
5689:
5688:
5680:
5675:
5674:
5673:
5671:
5670:
5664:
5659:
5654:
5629:
5623:
5617:
5615:
5614:
5609:
5596:
5590:
5588:
5586:
5585:
5580:
5578:
5577:
5572:
5571:
5563:
5552:
5550:
5549:
5544:
5533:
5532:
5527:
5526:
5511:
5505:
5503:
5501:
5500:
5495:
5474:
5471:
5469:
5468:
5463:
5435:
5433:
5432:
5427:
5414:
5408:
5406:
5404:
5403:
5398:
5396:
5395:
5390:
5389:
5381:
5370:
5368:
5367:
5362:
5351:
5350:
5345:
5344:
5329:
5220:
5218:
5217:
5212:
5206:
5192:
5191:
5183:
5175:
5152:
5151:
5146:
5145:
5137:
5122:
5120:
5119:
5114:
5112:
5098:
5097:
5089:
5081:
5061:
5060:
5055:
5054:
5046:
5031:
5029:
5028:
5023:
5021:
5004:
5003:
4998:
4997:
4989:
4969:
4968:
4963:
4962:
4954:
4944:
4940:
4936:
4891:
4886:
4863:
4859:
4814:
4809:
4786:
4782:
4779:
4776:
4768:
4764:
4690:
4685:
4657:
4656:
4648:
4633:
4631:
4630:
4625:
4590:
4589:
4560:
4558:
4557:
4552:
4540:
4538:
4537:
4532:
4529:
4518:
4517:
4480:
4475:
4447:
4446:
4438:
4413:
4411:
4410:
4405:
4378:
4376:
4375:
4374:
4353:
4352:
4337:
4336:
4326:
4318:
4315:
4310:
4279:
4277:
4276:
4271:
4253:
4251:
4250:
4245:
4227:
4225:
4224:
4219:
4207:
4205:
4204:
4203:
4182:
4181:
4166:
4165:
4155:
4147:
4132:
4127:
4049:
4048:
4032:
4027:
4017:
4012:
3991:is fixed. Then
3990:
3988:
3987:
3982:
3964:
3962:
3961:
3956:
3954:
3953:
3925:
3923:
3922:
3917:
3849:
3844:
3831:
3826:
3811:
3807:
3761:
3746:
3737:
3728:, in which case
3727:
3725:
3724:
3719:
3703:
3701:
3700:
3695:
3683:
3681:
3680:
3675:
3607:
3602:
3589:
3584:
3530:
3528:
3527:
3522:
3520:
3519:
3514:
3513:
3505:
3494:
3492:
3491:
3486:
3484:
3483:
3478:
3477:
3469:
3458:
3456:
3455:
3450:
3438:
3436:
3435:
3430:
3418:
3416:
3415:
3410:
3396:
3394:
3393:
3388:
3386:
3383:
3369:
3366:
3321:
3320:
3315:
3314:
3306:
3301:
3296:
3256:
3254:
3253:
3248:
3234:
3232:
3231:
3226:
3224:
3221:
3207:
3204:
3159:
3158:
3153:
3152:
3144:
3139:
3134:
3096:
3094:
3093:
3088:
3074:
3072:
3071:
3066:
3020:
3019:
3014:
3013:
3005:
3000:
2995:
2939:
2938:
2933:
2932:
2924:
2919:
2914:
2878:
2876:
2875:
2870:
2850:
2848:
2847:
2842:
2835:
2822:
2819:
2816:
2811:
2770:
2769:
2764:
2763:
2755:
2750:
2745:
2724:
2711:
2708:
2705:
2700:
2659:
2658:
2653:
2652:
2644:
2639:
2634:
2592:
2590:
2589:
2584:
2564:
2562:
2561:
2556:
2544:
2542:
2541:
2536:
2524:
2522:
2521:
2516:
2504:
2502:
2501:
2496:
2456:
2451:
2442:
2434:
2420:
2419:
2403:
2401:
2400:
2395:
2355:
2350:
2341:
2333:
2319:
2318:
2302:
2300:
2299:
2294:
2245:
2240:
2231:
2222:
2221:
2207:
2206:
2201:
2200:
2192:
2181:
2179:
2178:
2173:
2127:
2122:
2113:
2104:
2103:
2089:
2088:
2083:
2082:
2074:
2045:
2043:
2042:
2037:
2021:
2018:
2016:
2011:
1974:
1973:
1970:
1963:
1961:
1959:
1954:
1930:
1929:
1924:
1923:
1915:
1900:
1898:
1897:
1892:
1850:
1849:
1846:
1839:
1834:
1811:
1808:
1806:
1801:
1764:
1763:
1760:
1753:
1751:
1749:
1744:
1720:
1719:
1714:
1713:
1705:
1692:
1690:
1689:
1684:
1672:
1670:
1669:
1664:
1653:
1652:
1649:
1628:
1626:
1625:
1620:
1593:
1591:
1590:
1585:
1533:
1531:
1530:
1525:
1509:
1506:
1504:
1499:
1462:
1461:
1458:
1451:
1449:
1447:
1442:
1418:
1417:
1412:
1411:
1403:
1386:
1384:
1383:
1378:
1366:
1364:
1363:
1358:
1353:
1339:
1337:
1336:
1331:
1286:
1285:
1282:
1275:
1270:
1247:
1244:
1242:
1237:
1200:
1199:
1196:
1189:
1187:
1185:
1180:
1156:
1155:
1150:
1149:
1141:
1128:
1126:
1125:
1120:
1108:
1106:
1105:
1100:
1088:
1086:
1085:
1080:
1065:
1063:
1062:
1057:
1045:
1043:
1042:
1037:
1032:
1020:
1018:
1017:
1012:
1001:
1000:
997:
972:
961:
959:
958:
953:
951:
950:
949:
944:
943:
942:
923:
912:
902:
900:
886:
885:
880:
879:
871:
859:
857:
856:
851:
849:
848:
847:
846:
804:
802:
801:
796:
779:
778:
773:
772:
764:
748:
747:
742:
741:
733:
720:
718:
717:
712:
693:
691:
690:
685:
636:
631:
607:
606:
601:
600:
592:
577:
575:
574:
569:
539:
537:
536:
531:
507:
505:
504:
499:
487:
485:
484:
479:
477:
468:
448:
446:
445:
440:
426:
425:
420:
419:
411:
389:
388:
383:
382:
374:
361:
359:
358:
353:
322:
320:
319:
314:
298:
296:
295:
290:
275:
273:
272:
267:
218:
213:
189:
188:
183:
182:
174:
159:
157:
156:
151:
108:Fourier analysis
36:frequency domain
21:
18:Cosine transform
6031:
6030:
6026:
6025:
6024:
6022:
6021:
6020:
5996:
5995:
5994:
5985:
5981:
5972:
5970:
5966:
5955:
5950:
5949:
5945:
5936:
5934:
5918:
5917:
5913:
5898:
5878:
5877:
5873:
5863:
5862:
5858:
5845:Poincaré, Henri
5843:
5842:
5838:
5821:
5809:
5794:
5793:
5789:
5780:
5778:
5764:
5763:
5759:
5738:
5734:
5721:
5716:
5715:
5637:
5636:
5630:
5626:
5600:
5599:
5597:
5593:
5560:
5555:
5554:
5520:
5515:
5514:
5512:
5508:
5480:
5479:
5472:
5454:
5453:
5418:
5417:
5415:
5411:
5378:
5373:
5372:
5338:
5333:
5332:
5330:
5326:
5321:
5299:
5290:
5226:
5134:
5129:
5128:
5043:
5038:
5037:
5019:
5018:
4986:
4951:
4942:
4941:
4874:
4870:
4797:
4793:
4784:
4783:
4780:Euler's Formula
4775:
4708:
4704:
4667:
4637:
4636:
4578:
4570:
4569:
4567:Euler's formula
4543:
4542:
4527:
4526:
4494:
4457:
4427:
4426:
4419:
4366:
4344:
4328:
4327:
4319:
4282:
4281:
4256:
4255:
4230:
4229:
4195:
4173:
4157:
4156:
4148:
4034:
3993:
3992:
3967:
3966:
3939:
3934:
3933:
3932:is to insert a
3767:
3763:
3730:
3729:
3710:
3709:
3686:
3685:
3553:
3552:
3541:
3533:transform pairs
3502:
3497:
3496:
3466:
3461:
3460:
3441:
3440:
3421:
3420:
3401:
3400:
3303:
3265:
3264:
3239:
3238:
3141:
3103:
3102:
3079:
3078:
3002:
2921:
2883:
2882:
2861:
2860:
2857:
2855:Simplifications
2852:
2752:
2733:
2641:
2622:
2599:
2598:
2575:
2574:
2571:
2547:
2546:
2527:
2526:
2507:
2506:
2411:
2406:
2405:
2310:
2305:
2304:
2189:
2184:
2183:
2071:
2066:
2065:
2058:
1965:
1964:
1912:
1907:
1906:
1841:
1755:
1754:
1702:
1697:
1696:
1675:
1674:
1644:
1639:
1638:
1596:
1595:
1558:
1557:
1453:
1452:
1400:
1395:
1394:
1369:
1368:
1344:
1343:
1277:
1191:
1190:
1138:
1133:
1132:
1111:
1110:
1091:
1090:
1071:
1070:
1048:
1047:
1023:
1022:
992:
987:
986:
964:
962:
934:
924:
914:
868:
863:
862:
861:
838:
827:
822:
821:
810:
761:
730:
725:
724:
703:
702:
695:
589:
584:
583:
551:
550:
544:
513:
512:
510:normalized sinc
490:
489:
461:
460:
408:
371:
366:
365:
344:
343:
305:
304:
281:
280:
277:
171:
166:
165:
133:
132:
116:
88:complex numbers
28:
23:
22:
15:
12:
11:
5:
6029:
6027:
6019:
6018:
6013:
6008:
5998:
5997:
5993:
5992:
5979:
5943:
5911:
5896:
5886:(1927-01-02).
5871:
5856:
5836:
5807:
5787:
5757:
5731:
5730:
5729:
5720:
5717:
5714:
5713:
5701:
5698:
5695:
5692:
5686:
5683:
5669:
5663:
5653:
5650:
5647:
5644:
5624:
5607:
5591:
5576:
5569:
5566:
5542:
5539:
5536:
5531:
5525:
5506:
5493:
5490:
5487:
5461:
5425:
5409:
5394:
5387:
5384:
5360:
5357:
5354:
5349:
5343:
5323:
5322:
5320:
5317:
5316:
5315:
5310:
5305:
5298:
5295:
5289:
5286:
5225:
5222:
5210:
5205:
5201:
5198:
5195:
5189:
5186:
5179:
5174:
5171:
5167:
5164:
5161:
5158:
5155:
5150:
5143:
5140:
5111:
5107:
5104:
5101:
5095:
5092:
5085:
5080:
5077:
5073:
5070:
5067:
5064:
5059:
5052:
5049:
5017:
5013:
5010:
5007:
5002:
4995:
4992:
4984:
4981:
4978:
4975:
4972:
4967:
4960:
4957:
4950:
4947:
4945:
4943:
4939:
4935:
4932:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4890:
4885:
4882:
4878:
4873:
4869:
4866:
4862:
4858:
4855:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4813:
4808:
4805:
4801:
4796:
4792:
4789:
4787:
4785:
4777:
4774:
4771:
4767:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4738:
4735:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4711:
4707:
4703:
4700:
4697:
4694:
4689:
4684:
4681:
4677:
4673:
4670:
4668:
4666:
4663:
4660:
4654:
4651:
4645:
4644:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4588:
4585:
4581:
4577:
4565:. By applying
4550:
4525:
4522:
4516:
4513:
4510:
4507:
4504:
4501:
4497:
4493:
4490:
4487:
4484:
4479:
4474:
4471:
4467:
4463:
4460:
4458:
4456:
4453:
4450:
4444:
4441:
4435:
4434:
4418:
4415:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4373:
4369:
4365:
4362:
4359:
4356:
4351:
4347:
4343:
4340:
4335:
4331:
4325:
4322:
4314:
4309:
4306:
4302:
4298:
4295:
4292:
4289:
4269:
4266:
4263:
4243:
4240:
4237:
4217:
4214:
4211:
4202:
4198:
4194:
4191:
4188:
4185:
4180:
4176:
4172:
4169:
4164:
4160:
4154:
4151:
4145:
4142:
4139:
4136:
4131:
4126:
4123:
4119:
4115:
4112:
4109:
4105:
4102:
4099:
4096:
4092:
4089:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4047:
4044:
4041:
4037:
4031:
4026:
4022:
4016:
4011:
4008:
4004:
4000:
3980:
3977:
3974:
3952:
3949:
3946:
3942:
3915:
3912:
3909:
3905:
3902:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3848:
3843:
3840:
3836:
3830:
3825:
3821:
3817:
3814:
3810:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3766:
3760:
3757:
3754:
3750:
3743:
3740:
3717:
3693:
3673:
3670:
3667:
3663:
3660:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3606:
3601:
3598:
3594:
3588:
3583:
3580:
3576:
3572:
3569:
3566:
3563:
3560:
3540:
3537:
3518:
3511:
3508:
3482:
3475:
3472:
3448:
3428:
3408:
3384: is even.
3381:
3378:
3375:
3372:
3364:
3361:
3358:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3319:
3312:
3309:
3300:
3295:
3292:
3288:
3284:
3281:
3278:
3275:
3272:
3246:
3219:
3216:
3213:
3210:
3202:
3199:
3196:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3157:
3150:
3147:
3138:
3133:
3130:
3126:
3122:
3119:
3116:
3113:
3110:
3086:
3064:
3060:
3057:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3018:
3011:
3008:
2999:
2994:
2990:
2986:
2983:
2979:
2976:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2937:
2930:
2927:
2918:
2913:
2909:
2905:
2902:
2899:
2896:
2893:
2890:
2868:
2856:
2853:
2840:
2834:
2831:
2828:
2825:
2814:
2810:
2807:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2768:
2761:
2758:
2749:
2744:
2741:
2737:
2729:
2723:
2720:
2717:
2714:
2703:
2699:
2696:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2657:
2650:
2647:
2638:
2633:
2630:
2626:
2618:
2615:
2612:
2609:
2606:
2595:
2582:
2570:
2567:
2554:
2534:
2514:
2494:
2491:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2455:
2450:
2446:
2440:
2437:
2432:
2429:
2426:
2423:
2418:
2414:
2393:
2390:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2354:
2349:
2345:
2339:
2336:
2331:
2328:
2325:
2322:
2317:
2313:
2292:
2289:
2286:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2244:
2239:
2235:
2228:
2225:
2219:
2216:
2213:
2210:
2205:
2198:
2195:
2171:
2168:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2126:
2121:
2117:
2110:
2107:
2101:
2098:
2095:
2092:
2087:
2080:
2077:
2057:
2054:
2035:
2032:
2029:
2026:
2014:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1968:
1958:
1953:
1950:
1946:
1942:
1939:
1936:
1933:
1928:
1921:
1918:
1890:
1887:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1844:
1838:
1833:
1829:
1825:
1822:
1819:
1816:
1804:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1758:
1748:
1743:
1740:
1736:
1732:
1729:
1726:
1723:
1718:
1711:
1708:
1682:
1662:
1659:
1656:
1647:
1631:transform pair
1618:
1615:
1612:
1609:
1606:
1603:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1523:
1520:
1517:
1514:
1502:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1456:
1446:
1441:
1438:
1434:
1430:
1427:
1424:
1421:
1416:
1409:
1406:
1376:
1356:
1352:
1329:
1326:
1323:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1280:
1274:
1269:
1265:
1261:
1258:
1255:
1252:
1240:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1194:
1184:
1179:
1176:
1172:
1168:
1165:
1162:
1159:
1154:
1147:
1144:
1118:
1098:
1078:
1055:
1035:
1031:
1010:
1007:
1004:
995:
947:
941:
937:
933:
930:
927:
921:
917:
908:
905:
898:
895:
892:
889:
884:
877:
874:
845:
841:
837:
834:
830:
809:
806:
794:
791:
788:
785:
782:
777:
770:
767:
760:
757:
754:
751:
746:
739:
736:
710:
683:
680:
677:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
635:
630:
627:
623:
619:
616:
613:
610:
605:
598:
595:
580:
567:
564:
561:
558:
540:plotted above.
529:
526:
523:
520:
497:
474:
471:
438:
435:
432:
429:
424:
417:
414:
407:
404:
401:
398:
395:
392:
387:
380:
377:
351:
312:
288:
265:
262:
259:
255:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
217:
212:
209:
205:
201:
198:
195:
192:
187:
180:
177:
162:
149:
146:
143:
140:
115:
112:
96:Joseph Fourier
58:, the Fourier
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6028:
6017:
6014:
6012:
6009:
6007:
6004:
6003:
6001:
5989:
5983:
5980:
5965:
5961:
5954:
5947:
5944:
5932:
5928:
5927:
5922:
5915:
5912:
5907:
5903:
5899:
5897:0-521-06794-4
5893:
5889:
5885:
5881:
5875:
5872:
5867:
5860:
5857:
5852:
5851:
5846:
5840:
5837:
5832:
5826:
5818:
5814:
5810:
5804:
5800:
5799:
5791:
5788:
5777:on 2023-06-07
5776:
5772:
5768:
5761:
5758:
5755:
5754:0-471-04409-1
5751:
5747:
5746:
5741:
5736:
5733:
5727:
5723:
5722:
5718:
5699:
5693:
5684:
5681:
5648:
5642:
5634:
5628:
5625:
5621:
5605:
5595:
5592:
5574:
5564:
5537:
5529:
5510:
5507:
5491:
5488:
5485:
5477:
5459:
5451:
5447:
5443:
5439:
5423:
5413:
5410:
5392:
5382:
5355:
5347:
5328:
5325:
5318:
5314:
5311:
5309:
5306:
5304:
5301:
5300:
5296:
5294:
5287:
5285:
5283:
5279:
5276:may start by
5275:
5271:
5268:may start by
5267:
5262:
5257:
5255:
5251:
5243:
5239:
5235:
5234:phase-shifted
5230:
5224:Pros and cons
5223:
5221:
5208:
5196:
5184:
5165:
5162:
5156:
5148:
5138:
5126:
5102:
5090:
5071:
5065:
5057:
5047:
5035:
5015:
5008:
5000:
4990:
4982:
4979:
4973:
4965:
4955:
4948:
4946:
4937:
4933:
4930:
4923:
4920:
4917:
4914:
4908:
4905:
4899:
4893:
4880:
4876:
4871:
4867:
4864:
4860:
4856:
4853:
4846:
4843:
4840:
4837:
4831:
4828:
4822:
4816:
4803:
4799:
4794:
4790:
4788:
4772:
4769:
4765:
4758:
4755:
4752:
4749:
4743:
4740:
4736:
4733:
4727:
4724:
4721:
4718:
4712:
4709:
4705:
4698:
4692:
4679:
4675:
4671:
4669:
4661:
4649:
4621:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4586:
4583:
4579:
4568:
4564:
4548:
4523:
4520:
4514:
4511:
4508:
4505:
4502:
4499:
4495:
4488:
4482:
4469:
4465:
4461:
4459:
4451:
4439:
4424:
4416:
4414:
4401:
4395:
4389:
4386:
4383:
4380:
4371:
4363:
4360:
4357:
4349:
4345:
4341:
4338:
4333:
4329:
4323:
4320:
4304:
4300:
4293:
4287:
4267:
4264:
4261:
4241:
4235:
4215:
4212:
4209:
4200:
4192:
4189:
4186:
4178:
4174:
4170:
4167:
4162:
4158:
4152:
4149:
4140:
4134:
4121:
4117:
4113:
4110:
4107:
4100:
4094:
4090:
4087:
4077:
4074:
4071:
4065:
4062:
4059:
4053:
4050:
4045:
4042:
4039:
4035:
4024:
4020:
4006:
4002:
3998:
3978:
3975:
3972:
3950:
3947:
3944:
3940:
3931:
3926:
3913:
3910:
3907:
3903:
3900:
3890:
3887:
3884:
3878:
3875:
3872:
3866:
3863:
3857:
3851:
3838:
3834:
3823:
3819:
3815:
3812:
3808:
3801:
3798:
3795:
3789:
3786:
3780:
3777:
3774:
3768:
3764:
3758:
3752:
3741:
3738:
3715:
3707:
3691:
3671:
3668:
3665:
3661:
3658:
3648:
3645:
3642:
3636:
3633:
3630:
3624:
3621:
3615:
3609:
3596:
3592:
3578:
3574:
3570:
3564:
3558:
3550:
3546:
3538:
3536:
3534:
3516:
3506:
3480:
3470:
3459:swapped with
3446:
3426:
3419:swapped with
3406:
3397:
3376:
3370:
3362:
3359:
3356:
3349:
3346:
3343:
3340:
3334:
3331:
3325:
3317:
3307:
3290:
3286:
3282:
3276:
3270:
3262:
3260:
3259:even function
3244:
3235:
3222: is odd.
3214:
3208:
3200:
3197:
3194:
3187:
3184:
3181:
3178:
3172:
3169:
3163:
3155:
3145:
3128:
3124:
3120:
3114:
3108:
3100:
3084:
3075:
3062:
3058:
3055:
3048:
3045:
3042:
3039:
3033:
3030:
3024:
3016:
3006:
2992:
2988:
2984:
2981:
2977:
2974:
2967:
2964:
2961:
2958:
2952:
2949:
2943:
2935:
2925:
2911:
2907:
2903:
2900:
2894:
2888:
2880:
2866:
2854:
2851:
2838:
2829:
2823:
2812:
2808:
2805:
2798:
2795:
2792:
2789:
2783:
2780:
2774:
2766:
2756:
2739:
2735:
2727:
2718:
2712:
2701:
2697:
2694:
2687:
2684:
2681:
2678:
2672:
2669:
2663:
2655:
2645:
2628:
2624:
2616:
2610:
2604:
2594:
2580:
2568:
2566:
2552:
2532:
2512:
2492:
2489:
2482:
2479:
2473:
2470:
2464:
2458:
2448:
2444:
2438:
2435:
2430:
2424:
2416:
2412:
2391:
2388:
2381:
2378:
2372:
2369:
2363:
2357:
2347:
2343:
2337:
2334:
2329:
2323:
2315:
2311:
2290:
2287:
2284:
2277:
2274:
2271:
2268:
2262:
2259:
2253:
2247:
2237:
2233:
2226:
2223:
2217:
2211:
2203:
2193:
2169:
2166:
2159:
2156:
2153:
2150:
2144:
2141:
2135:
2129:
2119:
2115:
2108:
2105:
2099:
2093:
2085:
2075:
2063:
2055:
2053:
2051:
2046:
2033:
2030:
2027:
2024:
2012:
2005:
2002:
1999:
1996:
1990:
1987:
1984:
1978:
1966:
1948:
1944:
1940:
1934:
1926:
1916:
1904:
1901:
1888:
1885:
1878:
1875:
1872:
1869:
1863:
1860:
1854:
1842:
1831:
1827:
1823:
1820:
1817:
1814:
1802:
1795:
1792:
1789:
1786:
1780:
1777:
1774:
1768:
1756:
1738:
1734:
1730:
1724:
1716:
1706:
1694:
1680:
1657:
1645:
1632:
1613:
1610:
1604:
1601:
1581:
1575:
1572:
1566:
1563:
1555:
1551:
1547:
1543:
1538:
1534:
1521:
1518:
1515:
1512:
1500:
1493:
1490:
1487:
1484:
1478:
1475:
1472:
1466:
1454:
1436:
1432:
1428:
1422:
1414:
1404:
1392:
1390:
1350:
1340:
1327:
1324:
1321:
1314:
1311:
1308:
1305:
1299:
1296:
1290:
1278:
1267:
1263:
1259:
1256:
1253:
1250:
1238:
1231:
1228:
1225:
1222:
1216:
1213:
1210:
1204:
1192:
1174:
1170:
1166:
1160:
1152:
1142:
1130:
1116:
1076:
1068:
1029:
1005:
993:
984:
980:
971:
967:
945:
939:
931:
928:
919:
915:
906:
903:
896:
890:
882:
872:
843:
839:
835:
832:
828:
819:
814:
807:
805:
792:
786:
783:
775:
765:
758:
752:
744:
734:
722:
708:
700:
699:even function
694:
681:
678:
675:
668:
665:
662:
659:
653:
650:
644:
638:
625:
621:
617:
611:
603:
593:
579:
562:
556:
548:
524:
521:
511:
495:
472:
469:
458:
453:
449:
436:
430:
422:
412:
405:
402:
396:
393:
385:
375:
363:
349:
341:
336:
334:
330:
326:
310:
302:
286:
276:
263:
260:
257:
250:
247:
244:
241:
235:
232:
226:
220:
207:
203:
199:
193:
185:
175:
161:
144:
138:
130:
123:respectively.
120:
113:
111:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
72:odd component
69:
65:
61:
57:
49:
45:
41:
37:
32:
19:
5987:
5982:
5971:. Retrieved
5959:
5946:
5935:. Retrieved
5924:
5914:
5887:
5874:
5865:
5859:
5849:
5839:
5797:
5790:
5779:. Retrieved
5775:the original
5770:
5760:
5743:
5740:Mary L. Boas
5735:
5725:
5627:
5594:
5509:
5412:
5327:
5291:
5258:
5253:
5247:
5241:
5124:
4420:
3927:
3548:
3542:
3532:
3398:
3263:
3236:
3099:odd function
3076:
2881:
2858:
2597:
2572:
2059:
2047:
1905:
1902:
1695:
1636:
1630:
1393:
1341:
1131:
976:
969:
965:
723:
696:
582:
546:
543:
364:
340:odd function
337:
278:
164:
128:
126:
59:
53:
5553:instead of
5371:instead of
459:(of height
56:mathematics
6000:Categories
5973:2024-09-11
5937:2024-09-09
5926:IEEE Pulse
5781:2018-10-08
5719:References
3439:(and with
1548:about the
488:and width
114:Definition
104:statistics
78:concisely
68:sine waves
5825:cite book
5817:822959644
5694:ξ
5685:^
5662:⟷
5568:^
5492:ξ
5489:π
5460:ω
5424:ξ
5386:^
5197:ξ
5188:^
5166:−
5157:ξ
5142:^
5103:ξ
5094:^
5066:ξ
5051:^
5009:ξ
4994:^
4980:−
4974:ξ
4959:^
4921:ξ
4918:π
4909:
4889:∞
4884:∞
4881:−
4877:∫
4865:−
4844:ξ
4841:π
4832:
4812:∞
4807:∞
4804:−
4800:∫
4756:ξ
4753:π
4744:
4734:−
4725:ξ
4722:π
4713:
4688:∞
4683:∞
4680:−
4676:∫
4662:ξ
4653:^
4613:
4598:
4512:ξ
4506:π
4500:−
4478:∞
4473:∞
4470:−
4466:∫
4452:ξ
4443:^
4361:−
4346:π
4330:δ
4324:δ
4313:∞
4308:∞
4305:−
4301:∫
4239:→
4236:δ
4228:Now when
4190:−
4175:π
4159:δ
4153:δ
4130:∞
4125:∞
4122:−
4118:∫
4091:ξ
4075:−
4066:ξ
4063:π
4054:
4046:ξ
4043:δ
4040:−
4030:∞
4021:∫
4015:∞
4010:∞
4007:−
4003:∫
3973:δ
3951:ξ
3948:δ
3945:−
3911:ξ
3888:−
3879:ξ
3876:π
3867:
3847:∞
3842:∞
3839:−
3835:∫
3829:∞
3820:∫
3799:−
3756:→
3669:ξ
3646:−
3637:ξ
3634:π
3625:
3605:∞
3600:∞
3597:−
3593:∫
3587:∞
3582:∞
3579:−
3575:∫
3510:^
3474:^
3427:ξ
3360:ξ
3347:ξ
3344:π
3335:
3326:ξ
3311:^
3299:∞
3294:∞
3291:−
3287:∫
3198:ξ
3185:ξ
3182:π
3173:
3164:ξ
3149:^
3137:∞
3132:∞
3129:−
3125:∫
3077:Also, if
3059:ξ
3046:ξ
3043:π
3034:
3025:ξ
3010:^
2998:∞
2989:∫
2978:ξ
2965:ξ
2962:π
2953:
2944:ξ
2929:^
2917:∞
2908:∫
2867:ξ
2813:⏟
2809:ξ
2796:ξ
2793:π
2784:
2775:ξ
2760:^
2748:∞
2743:∞
2740:−
2736:∫
2702:⏟
2698:ξ
2685:ξ
2682:π
2673:
2664:ξ
2649:^
2637:∞
2632:∞
2629:−
2625:∫
2513:α
2480:α
2474:
2454:∞
2445:∫
2439:π
2425:α
2379:α
2373:
2353:∞
2344:∫
2338:π
2324:α
2275:ξ
2272:π
2263:
2243:∞
2234:∫
2227:π
2212:ξ
2197:^
2157:ξ
2154:π
2145:
2125:∞
2116:∫
2109:π
2094:ξ
2079:^
2013:⏞
2003:ξ
2000:π
1991:
1985:⋅
1957:∞
1952:∞
1949:−
1945:∫
1935:ξ
1920:^
1876:ξ
1873:π
1864:
1837:∞
1828:∫
1803:⏞
1793:ξ
1790:π
1781:
1775:⋅
1747:∞
1742:∞
1739:−
1735:∫
1725:ξ
1710:^
1614:ξ
1605:
1576:ξ
1567:
1501:⏞
1491:ξ
1488:π
1479:
1473:⋅
1445:∞
1440:∞
1437:−
1433:∫
1423:ξ
1408:^
1375:∞
1355:∞
1351:−
1312:ξ
1309:π
1300:
1273:∞
1264:∫
1239:⏞
1229:ξ
1226:π
1217:
1211:⋅
1183:∞
1178:∞
1175:−
1171:∫
1161:ξ
1146:^
1097:∞
1054:∞
1034:∞
1030:−
946:α
932:ξ
929:π
920:−
907:α
904:π
891:ξ
876:^
836:α
833:−
787:ξ
784:−
769:^
753:ξ
738:^
709:ξ
666:ξ
663:π
654:
634:∞
629:∞
626:−
622:∫
612:ξ
597:^
525:ξ
508:) is the
431:ξ
416:^
406:−
397:ξ
394:−
379:^
350:ξ
325:frequency
311:ξ
248:ξ
245:π
236:
216:∞
211:∞
208:−
204:∫
194:ξ
179:^
5964:Archived
5931:Archived
5847:(1895).
5440:and the
5297:See also
5242:together
5125:negative
329:position
80:contains
5478:equals
5450:radians
4561:is the
1546:degrees
1542:rotated
303:, then
50:domain.
48:spatial
5904:
5894:
5815:
5805:
5752:
5676:
5655:
5473:
5442:second
4541:where
3930:Cauchy
3545:cosine
3257:is an
3097:is an
2505:using
1550:origin
299:means
5967:(PDF)
5956:(PDF)
5438:Hertz
5319:Notes
5250:phase
1069:from
5902:ISBN
5892:ISBN
5831:link
5813:OCLC
5803:ISBN
5750:ISBN
3976:>
1971:even
1544:180
1283:even
1197:even
998:even
977:The
578:is:
545:The
331:and
301:time
160:is:
127:The
102:and
62:are
44:time
5254:and
4906:sin
4829:cos
4741:sin
4710:cos
4610:sin
4595:cos
4051:cos
3864:cos
3749:lim
3622:cos
3495:or
3332:cos
3170:sin
3031:cos
2950:sin
2781:cos
2670:sin
2471:sin
2370:cos
2260:sin
2142:cos
1988:sin
1861:sin
1847:odd
1778:sin
1761:odd
1650:odd
1602:sin
1564:sin
1476:cos
1459:odd
1387:of
1367:to
1297:cos
1214:cos
1089:to
1046:to
651:cos
549:of
335:).
323:is
279:If
233:sin
131:of
110:.
90:or
54:In
46:or
6002::
5962:.
5958:.
5929:.
5923:.
5900:.
5882:;
5827:}}
5823:{{
5811:.
5769:.
5742:,
5284:.
3551::
3535:.
2034:0.
1693::
1522:0.
1129::
721::
362::
5976:.
5940:.
5908:.
5833:)
5819:.
5784:.
5700:.
5697:)
5691:(
5682:f
5668:F
5652:)
5649:t
5646:(
5643:f
5622:.
5606:f
5589:.
5575:c
5565:f
5541:)
5538:f
5535:(
5530:c
5524:F
5504:.
5486:2
5407:.
5393:s
5383:f
5359:)
5356:f
5353:(
5348:s
5342:F
5209:.
5204:]
5200:)
5194:(
5185:f
5178:[
5173:m
5170:I
5163:=
5160:)
5154:(
5149:s
5139:f
5110:]
5106:)
5100:(
5091:f
5084:[
5079:e
5076:R
5072:=
5069:)
5063:(
5058:c
5048:f
5016:.
5012:)
5006:(
5001:s
4991:f
4983:i
4977:)
4971:(
4966:c
4956:f
4949:=
4938:)
4934:t
4931:d
4927:)
4924:t
4915:2
4912:(
4903:)
4900:t
4897:(
4894:f
4872:(
4868:i
4861:)
4857:t
4854:d
4850:)
4847:t
4838:2
4835:(
4826:)
4823:t
4820:(
4817:f
4795:(
4791:=
4773:t
4770:d
4766:)
4762:)
4759:t
4750:2
4747:(
4737:i
4731:)
4728:t
4719:2
4716:(
4706:(
4702:)
4699:t
4696:(
4693:f
4672:=
4665:)
4659:(
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4622:,
4619:)
4616:x
4607:i
4604:+
4601:x
4592:=
4587:x
4584:i
4580:e
4576:(
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4503:2
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4489:t
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4462:=
4455:)
4449:(
4440:f
4402:.
4399:)
4396:t
4393:(
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4387:=
4384:x
4381:d
4372:2
4368:)
4364:t
4358:x
4355:(
4350:2
4342:4
4339:+
4334:2
4321:2
4297:)
4294:t
4291:(
4288:f
4268:t
4265:=
4262:x
4242:0
4216:.
4213:x
4210:d
4201:2
4197:)
4193:t
4187:x
4184:(
4179:2
4171:4
4168:+
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4150:2
4144:)
4141:x
4138:(
4135:f
4114:=
4111:x
4108:d
4104:)
4101:x
4098:(
4095:f
4088:d
4084:)
4081:)
4078:t
4072:x
4069:(
4060:2
4057:(
4036:e
4025:0
3999:2
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3941:e
3914:.
3908:d
3904:x
3901:d
3897:)
3894:)
3891:t
3885:x
3882:(
3873:2
3870:(
3861:)
3858:x
3855:(
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3824:0
3816:2
3813:=
3809:)
3805:)
3802:h
3796:t
3793:(
3790:f
3787:+
3784:)
3781:h
3778:+
3775:t
3772:(
3769:f
3765:(
3759:0
3753:h
3742:2
3739:1
3716:t
3692:f
3672:.
3666:d
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3659:d
3655:)
3652:)
3649:t
3643:x
3640:(
3631:2
3628:(
3619:)
3616:x
3613:(
3610:f
3571:=
3568:)
3565:t
3562:(
3559:f
3517:c
3507:f
3481:s
3471:f
3447:f
3407:t
3380:)
3377:t
3374:(
3371:f
3363:,
3357:d
3353:)
3350:t
3341:2
3338:(
3329:)
3323:(
3318:c
3308:f
3283:=
3280:)
3277:t
3274:(
3271:f
3245:f
3218:)
3215:t
3212:(
3209:f
3201:,
3195:d
3191:)
3188:t
3179:2
3176:(
3167:)
3161:(
3156:s
3146:f
3121:=
3118:)
3115:t
3112:(
3109:f
3085:f
3063:.
3056:d
3052:)
3049:t
3040:2
3037:(
3028:)
3022:(
3017:c
3007:f
2993:0
2985:2
2982:+
2975:d
2971:)
2968:t
2959:2
2956:(
2947:)
2941:(
2936:s
2926:f
2912:0
2904:2
2901:=
2898:)
2895:t
2892:(
2889:f
2839:.
2833:)
2830:t
2827:(
2824:f
2806:d
2802:)
2799:t
2790:2
2787:(
2778:)
2772:(
2767:c
2757:f
2728:+
2722:)
2719:t
2716:(
2713:f
2695:d
2691:)
2688:t
2679:2
2676:(
2667:)
2661:(
2656:s
2646:f
2617:=
2614:)
2611:t
2608:(
2605:f
2581:f
2553:x
2533:t
2493:x
2490:d
2486:)
2483:x
2477:(
2468:)
2465:x
2462:(
2459:f
2449:0
2436:2
2431:=
2428:)
2422:(
2417:s
2413:F
2392:x
2389:d
2385:)
2382:x
2376:(
2367:)
2364:x
2361:(
2358:f
2348:0
2335:2
2330:=
2327:)
2321:(
2316:c
2312:F
2291:.
2288:t
2285:d
2281:)
2278:t
2269:2
2266:(
2257:)
2254:t
2251:(
2248:f
2238:0
2224:2
2218:=
2215:)
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2204:s
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2167:d
2163:)
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2148:(
2139:)
2136:t
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2100:=
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2091:(
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2031:=
2028:t
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2009:)
2006:t
1997:2
1994:(
1982:)
1979:t
1976:(
1967:f
1941:=
1938:)
1932:(
1927:s
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1886:d
1882:)
1879:t
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1867:(
1858:)
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1821:=
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1784:(
1772:)
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1766:(
1757:f
1731:=
1728:)
1722:(
1717:s
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1646:f
1633:.
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1611:a
1608:(
1582:.
1579:)
1573:a
1570:(
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1516:t
1513:d
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1494:t
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1482:(
1470:)
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1429:=
1426:)
1420:(
1415:c
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1328:.
1325:t
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1303:(
1294:)
1291:t
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1268:0
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1257:=
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1220:(
1208:)
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1167:=
1164:)
1158:(
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1077:0
1009:)
1006:t
1003:(
994:f
970:π
968:=
966:α
940:2
936:)
926:(
916:e
897:=
894:)
888:(
883:c
873:f
844:2
840:t
829:e
793:.
790:)
781:(
776:c
766:f
759:=
756:)
750:(
745:c
735:f
682:.
679:t
676:d
672:)
669:t
660:2
657:(
648:)
645:t
642:(
639:f
618:=
615:)
609:(
604:c
594:f
566:)
563:t
560:(
557:f
528:)
522:a
519:(
496:a
473:a
470:1
437:.
434:)
428:(
423:s
413:f
403:=
400:)
391:(
386:s
376:f
287:t
264:.
261:t
258:d
254:)
251:t
242:2
239:(
230:)
227:t
224:(
221:f
200:=
197:)
191:(
186:s
176:f
148:)
145:t
142:(
139:f
20:)
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