20:
1526:
5218:
5019:
4627:
2838:
5014:{\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}}}
108:
802:
1327:
441:
4215:
1888:
2589:
5282:
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.
5281:
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
5252:
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
3913:
4623:
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
3062:
1123:
1687:
2033:
1521:
3983:
3384:
3222:
4528:
2833:{\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)}\,.}
3671:
949:
4401:
2290:
2169:
111:
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,
3720:
2873:
1897:
5699:
1385:
2492:
2391:
681:
263:
3255:
3093:
4417:
1322:{\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.}
436:
792:
4210:{\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.}
1883:{\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}
5208:
5110:
3543:
853:
4632:
4422:
4272:
2174:
2056:
5540:
5358:
4621:
1008:
1660:
847:
3952:
2396:
2295:
5245:
amplitude information inside its complex valued result. But a disadvantage is its requirement on understanding complex numbers, complex exponentials, and negative frequency.
475:
5877:
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
5576:
5394:
3518:
3482:
4241:
3978:
1581:
1616:
1354:
1033:
5491:
5248:
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
5459:
2512:
1374:
1096:
1053:
527:
5919:
5627:
5423:
3426:
2866:
708:
565:
349:
310:
147:
5819:
4267:
5733:
5605:
4548:
3715:
3691:
3446:
3406:
3244:
3084:
2580:
2552:
2532:
1680:
1116:
1076:
495:
286:
3520:). 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
574:
156:
3908:{\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 .}
3057:{\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 \,.}
3917:
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
2028:{\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.}
356:
1516:{\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.}
715:
1525:
5119:
3388:
Remarkably, these last two simplified inversion formulas look identical to the original sine and cosine transforms, respectively, though with
5894:
5795:
5028:
19:
5301:
5884:
5742:
5755:
3379:{\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.}}}
3217:{\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.}}}
5952:
5226:
5237:
An advantage of the modern
Fourier transform is that while the sine and cosine transforms together are required to extract the
107:
801:
6004:
5909:
4523:{\displaystyle {\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\xi t}\,dt\\\end{aligned}}\,}
5994:
2868:, the concept of negative frequency can be avoided by doubling the result of integrating over non-negative frequencies:
5624:
has this symmetry even when the original functions aren't even or odd. A notation to denote
Fourier transform pairs is
4551:
5270:
5999:
5608:
5291:
5254:
498:
71:
both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of
5607:
and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see
5217:
3666:{\displaystyle f(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x)\cos(2\pi \xi (x-t))\,dx\,d\xi .}
2582:
can be recovered from its sine and cosine transforms under the usual hypotheses using the inversion formula:
944:{\displaystyle {\hat {f}}^{c}(\xi )={\sqrt {\tfrac {\pi }{\alpha }}}\,e^{-{\frac {(\pi \xi )^{2}}{\alpha }}}}
5868:
5296:
5266:
5262:
5258:
5227:
same frequency, but whose amplitude and phase depends on the amplitudes of the original sine and cosine wave
5505:
5323:
2049:
Just like the
Fourier transform takes the form of different equations with different constant factors (see
5021:
Because of this relationship, the cosine transform of functions whose
Fourier transform is known (e.g. in
4560:
4396:{\displaystyle f(t)\int _{-\infty }^{\infty }{\frac {2\delta }{\delta ^{2}+4\pi ^{2}(x-t)^{2}}}\,dx=f(t).}
2285:{\displaystyle {\hat {f}}^{s}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\sin(2\pi \xi t)\,dt.}
2038:
1530:
1377:
1055:
971:
970:
shown in the overbraces in the following equations dramatically simplify the integrands when transforming
967:
60:
2164:{\displaystyle {\hat {f}}^{c}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\cos(2\pi \xi t)\,dt}
5872:
977:
1629:
812:
3924:
1542:
1538:
451:
445:
5545:
5363:
3487:
3451:
317:
4220:
3957:
1548:
809:
is a mirror image of its right half and its sine transform is entirely 0. Gaussians have the form
5813:
5464:
5249:
1618:
is the above plot. Thus, the sine wave function and the time-shifted Dirac delta function form a
1586:
80:
52:
4555:
1334:
1013:
72:
5470:
5890:
5880:
5833:
5801:
5791:
5738:
5621:
5434:
5022:
4411:
3694:
2554:
is often instead used to represent a spatial domain when transforming to spatial frequencies.
2050:
806:
321:
88:
64:
63:
of the function plus cosine waves representing the even component of the function. The modern
5694:{\displaystyle f(t)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi ).}
5444:
2497:
1359:
1081:
1038:
503:
5837:
5241:
information of a frequency, the modern
Fourier transform instead compactly packs both phase
96:
24:
316:
in cycles per unit time, but in the abstract, they can be any dual pair of variables (e.g.
5408:
3411:
2851:
1541:. Their cosine transform is entirely zero. The above odd function contains two half-sized
693:
541:
334:
295:
123:
5763:
4246:
2487:{\displaystyle F_{s}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\sin(\alpha x)\,dx}
2386:{\displaystyle F_{c}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\cos(\alpha x)\,dx}
676:{\displaystyle {\hat {f}}^{c}(\xi )=\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt.}
258:{\displaystyle {\hat {f}}^{s}(\xi )=\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt.}
5590:
4533:
3700:
3676:
3431:
3391:
3229:
3069:
2565:
2537:
2517:
1665:
1534:
1101:
1061:
480:
271:
84:
76:
5941:
5988:
5238:
5222:
3247:
1010:. Since cosine is an even function and because the integral of an even function from
687:
5728:
3087:
328:
5221:
Adding a sine wave (red) and a cosine wave (blue) of the same frequency results a
2051:
Fourier transform § Unitarity and definition for square integrable functions
5229:. Hence, at a particular frequency, the sine transform and the cosine transform
1098:, the cosine transform of any even function can be simplified to avoid negative
44:
5914:
92:
56:
5805:
313:
28:
5979:, Journal of computational and applied mathematics 112.1-2 (1999): 229-241.
5785:
5233:
essentially only represent one sine wave that could have any phase shift.
5879:(4th ed.). Cambridge, UK: Cambridge University Press. p. 189.
3250:, then the sine transform is zero, so its inversion also simplifies to:
974:. Some authors even only define the cosine transform for even functions
5438:
5025:) can be simply found by taking the real part of the Fourier transform:
1626:
Similarly, because sin is odd, the sine transform of any odd function
5430:
3918:
3533:
2041:, while the cosine transform represents the even part of a function.
3090:, then the cosine transform is zero, so its inversion simplifies to:
2053:
for discussion), other authors also define the cosine transform as
440:
5426:
5216:
1524:
800:
439:
106:
36:
18:
289:
87:'s original transform equations and are still preferred in some
32:
5023:
Fourier transform § Tables of important Fourier transforms
5977:
A robust double exponential formula for Fourier-type integrals
3673:
This theorem is often stated under different hypotheses, that
5787:
An Introduction to Partial Differential Equations with MATLAB
5656:
5512:
5330:
1892:
and the sine transform of any even function is simply zero:
431:{\displaystyle {\hat {f}}^{s}(-\xi )=-{\hat {f}}^{s}(\xi ).}
95:
applications and may be better suited as an introduction to
5717:, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211
1380:, the cosine transform of any odd function is simply zero:
787:{\displaystyle {\hat {f}}^{c}(\xi )={\hat {f}}^{c}(-\xi ).}
5433:
as units, these transforms are sometimes expressed using
23:
The sine and cosine transforms convert a function into a
5203:{\displaystyle {\hat {f}}^{s}(\xi )=-\mathrm {Im} {}\,.}
2585:
Fourier inversion (from the sine and cosine transforms)
68:
4563:
3725:
3536:, the full inversion formula can also be rewritten as
2848:
Note that since both integrands are even functions of
890:
456:
5630:
5593:
5548:
5508:
5473:
5447:
5411:
5366:
5326:
5122:
5031:
4630:
4536:
4420:
4275:
4249:
4223:
3986:
3960:
3927:
3723:
3703:
3679:
3546:
3490:
3454:
3434:
3414:
3394:
3258:
3232:
3096:
3072:
2876:
2854:
2592:
2568:
2540:
2520:
2500:
2399:
2298:
2177:
2059:
1900:
1690:
1668:
1632:
1589:
1551:
1388:
1362:
1337:
1126:
1104:
1084:
1064:
1041:
1016:
980:
856:
815:
718:
696:
577:
544:
506:
483:
454:
359:
337:
298:
274:
159:
126:
5910:"Highlights in the History of the Fourier Transform"
5269:
of its input, to avoid having to compute the entire
5105:{\displaystyle {\hat {f}}^{c}(\xi )=\mathrm {Re} {}}
2292:Another convention defines the cosine transform as
5855:Introduction to the theory of the Fourier integral
5839:Theorie analytique de la propagation de la chaleur
5693:
5599:
5570:
5534:
5485:
5453:
5425:in cycles per unit time, which typically uses the
5417:
5388:
5352:
5202:
5104:
5013:
4615:
4542:
4522:
4395:
4261:
4235:
4209:
3972:
3946:
3907:
3709:
3685:
3665:
3512:
3476:
3440:
3420:
3400:
3378:
3238:
3216:
3078:
3056:
2860:
2832:
2574:
2546:
2526:
2506:
2486:
2385:
2284:
2163:
2027:
1882:
1674:
1654:
1610:
1575:
1515:
1368:
1348:
1321:
1110:
1090:
1070:
1047:
1027:
1002:
943:
841:
786:
702:
675:
559:
521:
489:
469:
430:
343:
304:
280:
257:
141:
55:that decompose arbitrary functions into a sum of
3737:
2534:is typically used to represent the time domain,
5756:"Fourier Transform, Cosine and Sine Transforms"
5502:The cosine transform is sometimes denoted with
5405:While this article uses ordinary frequency for
5116:of the imaginary part of the Fourier transform:
968:multiplication rules for even and odd functions
952:also is a Gaussian. The plotted Gaussian uses
5734:Mathematical Methods in the Physical Sciences
5320:The sine transform is sometimes denoted with
8:
5790:(Second ed.). Boca Raton. p. 221.
805:Like all even functions, the left half of a
5737:, 2nd Ed, John Wiley & Sons Inc, 1983.
31:. The inverse transform converts back to a
5857:, Oxford at the Clarendon Press, p. 1
5818:: CS1 maint: location missing publisher (
5191:
5169:
5097:
5075:
2514:as the transformation variable. And while
5668:
5667:
5655:
5654:
5649:
5647:
5646:
5629:
5592:
5562:
5551:
5550:
5547:
5517:
5511:
5510:
5507:
5472:
5446:
5410:
5380:
5369:
5368:
5365:
5335:
5329:
5328:
5325:
5196:
5171:
5170:
5165:
5157:
5136:
5125:
5124:
5121:
5077:
5076:
5071:
5063:
5045:
5034:
5033:
5030:
5003:
4988:
4977:
4976:
4974:
4953:
4942:
4941:
4918:
4876:
4868:
4841:
4799:
4791:
4767:
4728:
4675:
4667:
4636:
4635:
4631:
4629:
4571:
4562:
4535:
4519:
4508:
4487:
4465:
4457:
4426:
4425:
4421:
4419:
4368:
4359:
4337:
4321:
4306:
4300:
4292:
4274:
4248:
4222:
4197:
4188:
4166:
4150:
4135:
4117:
4109:
4095:
4082:
4075:
4027:
4017:
4012:
4002:
3994:
3985:
3959:
3932:
3926:
3895:
3888:
3834:
3826:
3816:
3811:
3740:
3724:
3722:
3702:
3697:on an open interval containing the point
3678:
3653:
3646:
3592:
3584:
3574:
3566:
3545:
3504:
3493:
3492:
3489:
3468:
3457:
3456:
3453:
3433:
3413:
3393:
3371:
3354:
3344:
3305:
3294:
3293:
3286:
3278:
3257:
3231:
3209:
3192:
3182:
3143:
3132:
3131:
3124:
3116:
3095:
3071:
3050:
3043:
3004:
2993:
2992:
2985:
2980:
2969:
2962:
2923:
2912:
2911:
2904:
2899:
2875:
2853:
2826:
2807:
2806:
2793:
2754:
2743:
2742:
2735:
2727:
2720:
2715:
2696:
2695:
2682:
2643:
2632:
2631:
2624:
2616:
2609:
2591:
2567:
2539:
2519:
2499:
2477:
2441:
2436:
2422:
2404:
2398:
2376:
2340:
2335:
2321:
2303:
2297:
2272:
2230:
2225:
2209:
2191:
2180:
2179:
2176:
2154:
2112:
2107:
2091:
2073:
2062:
2061:
2058:
2012:
2006:
1958:
1951:
1944:
1936:
1914:
1903:
1902:
1899:
1873:
1834:
1824:
1819:
1802:
1796:
1748:
1741:
1734:
1726:
1704:
1693:
1692:
1689:
1667:
1637:
1631:
1588:
1550:
1500:
1494:
1446:
1439:
1432:
1424:
1402:
1391:
1390:
1387:
1361:
1338:
1336:
1309:
1270:
1260:
1255:
1238:
1232:
1184:
1177:
1170:
1162:
1140:
1129:
1128:
1125:
1103:
1083:
1063:
1040:
1017:
1015:
985:
979:
927:
911:
907:
902:
888:
870:
859:
858:
855:
831:
820:
814:
763:
752:
751:
732:
721:
720:
717:
695:
663:
621:
613:
591:
580:
579:
576:
543:
505:
482:
455:
453:
410:
399:
398:
373:
362:
361:
358:
336:
297:
273:
245:
203:
195:
173:
162:
161:
158:
125:
4243:, the integrand tends to zero except at
5721:
5313:
5112:while the sine transform is simply the
686:The cosine transform is necessarily an
5811:
5713:Whittaker, Edmund, and James Watson,
5535:{\displaystyle {\mathcal {F}}_{c}(f)}
5353:{\displaystyle {\mathcal {F}}_{s}(f)}
4616:{\textstyle (e^{ix}=\cos x+i\sin x),}
327:The sine transform is necessarily an
7:
4410:The complex exponential form of the
5908:Valentinuzzi, Max E. (2016-01-25).
3226:Likewise, if the original function
5842:. Paris: G. Carré. pp. 108ff.
5302:List of Fourier-related transforms
5161:
5158:
5067:
5064:
4877:
4872:
4800:
4795:
4676:
4671:
4466:
4461:
4406:Relation with complex exponentials
4301:
4296:
4118:
4113:
4018:
4003:
3998:
3835:
3830:
3817:
3593:
3588:
3575:
3570:
3287:
3282:
3125:
3120:
2986:
2905:
2736:
2731:
2625:
2620:
2442:
2341:
2231:
2113:
2037:The sine transform represents the
1945:
1940:
1825:
1735:
1730:
1662:also simplifies to avoid negative
1543:time-shifted Dirac delta functions
1433:
1428:
1363:
1343:
1261:
1171:
1166:
1085:
1042:
1022:
1003:{\displaystyle f_{\text{even}}(t)}
622:
617:
204:
199:
83:, they more closely correspond to
14:
5940:Williams, Lance R. (2011-09-06).
1655:{\displaystyle f_{\text{odd}}(t)}
842:{\displaystyle e^{-\alpha t^{2}}}
444:The cosine transform of a simple
4269:, so that formally the above is
3947:{\displaystyle e^{-\delta \xi }}
1583:Likewise, the sine transform of
962:and is its own cosine transform.
5958:from the original on 2024-05-02
5922:from the original on 2024-05-15
3532:Using the addition formula for
1545:. Its sine transform is simply
1529:Odd functions are unchanged if
470:{\displaystyle {\tfrac {1}{a}}}
5685:
5679:
5650:
5640:
5634:
5587:The usual hypotheses are that
5571:{\displaystyle {\hat {f}}^{c}}
5556:
5529:
5523:
5389:{\displaystyle {\hat {f}}^{s}}
5374:
5347:
5341:
5192:
5188:
5182:
5176:
5166:
5148:
5142:
5130:
5098:
5094:
5088:
5082:
5072:
5057:
5051:
5039:
5000:
4994:
4982:
4965:
4959:
4947:
4915:
4900:
4891:
4885:
4838:
4823:
4814:
4808:
4750:
4735:
4719:
4704:
4690:
4684:
4653:
4647:
4641:
4607:
4564:
4480:
4474:
4443:
4437:
4431:
4387:
4381:
4356:
4343:
4285:
4279:
4227:
4185:
4172:
4132:
4126:
4092:
4086:
4072:
4069:
4057:
4045:
3885:
3882:
3870:
3858:
3849:
3843:
3793:
3781:
3772:
3760:
3744:
3643:
3640:
3628:
3616:
3607:
3601:
3556:
3550:
3513:{\displaystyle {\hat {f}}^{c}}
3498:
3477:{\displaystyle {\hat {f}}^{s}}
3462:
3368:
3362:
3341:
3326:
3317:
3311:
3299:
3268:
3262:
3206:
3200:
3179:
3164:
3155:
3149:
3137:
3106:
3100:
3040:
3025:
3016:
3010:
2998:
2959:
2944:
2935:
2929:
2917:
2886:
2880:
2821:
2815:
2790:
2775:
2766:
2760:
2748:
2710:
2704:
2679:
2664:
2655:
2649:
2637:
2602:
2596:
2474:
2465:
2456:
2450:
2416:
2410:
2373:
2364:
2355:
2349:
2315:
2309:
2269:
2254:
2245:
2239:
2203:
2197:
2185:
2151:
2136:
2127:
2121:
2085:
2079:
2067:
1997:
1982:
1970:
1964:
1926:
1920:
1908:
1870:
1855:
1846:
1840:
1787:
1772:
1760:
1754:
1716:
1710:
1698:
1649:
1643:
1605:
1596:
1567:
1558:
1485:
1470:
1458:
1452:
1414:
1408:
1396:
1331:And because the integral from
1306:
1291:
1282:
1276:
1223:
1208:
1196:
1190:
1152:
1146:
1134:
997:
991:
924:
914:
882:
876:
864:
778:
769:
757:
744:
738:
726:
660:
645:
636:
630:
603:
597:
585:
554:
548:
516:
507:
422:
416:
404:
388:
379:
367:
242:
227:
218:
212:
185:
179:
167:
136:
130:
1:
5975:Takuya Ooura, Masatake Mori,
1378:any odd function from is zero
5784:Coleman, Matthew P. (2013).
4236:{\displaystyle \delta \to 0}
3973:{\displaystyle \delta >0}
1576:{\displaystyle \sin(a\xi ).}
849:and their cosine transform:
5715:A Course in Modern Analysis
4552:square root of negative one
3528:Overview of inversion proof
1611:{\displaystyle \sin(a\xi )}
797:Odd and even simplification
690:of frequency, i.e. for all
331:of frequency, i.e. for all
27:representation as a sum of
6021:
5271:discrete Fourier transform
5259:assuming an even extension
5225:sine wave (green) of that
3538:Fourier's integral formula
2393:and the sine transform as
2171:and the sine transform as
1349:{\displaystyle {-}\infty }
1028:{\displaystyle {-}\infty }
49:sine and cosine transforms
16:Variant Fourier transforms
5949:www.cs.unm.edu/~williams/
5853:Edwin Titchmarsh (1948),
5609:Fourier inversion theorem
5486:{\displaystyle 2\pi \xi }
5292:Discrete cosine transform
5267:assuming an odd extension
5255:discrete cosine transform
4414:used more often today is
3954:into the integral, where
3693:is integrable, and is of
570:Fourier cosine transform
5942:"Even and odd functions"
5869:Whittaker, Edmund Taylor
5754:Nyack, Cuthbert (1996).
5620:The more general modern
536:Fourier cosine transform
5454:{\displaystyle \omega }
5441:) per unit time, where
5437:in angular units (e.g.
5297:Discrete sine transform
5263:discrete sine transform
2809:even component of
2507:{\displaystyle \alpha }
1369:{\displaystyle \infty }
1091:{\displaystyle \infty }
1048:{\displaystyle \infty }
522:{\displaystyle (a\xi )}
152:Fourier sine transform
5873:Watson, George Neville
5695:
5601:
5572:
5536:
5487:
5455:
5419:
5390:
5354:
5234:
5204:
5106:
5015:
4617:
4544:
4524:
4397:
4263:
4237:
4211:
3974:
3948:
3909:
3711:
3687:
3667:
3514:
3478:
3442:
3422:
3402:
3380:
3240:
3218:
3080:
3058:
2862:
2834:
2698:odd component of
2576:
2562:The original function
2548:
2528:
2508:
2488:
2387:
2286:
2165:
2039:odd part of a function
2029:
1884:
1676:
1656:
1623:
1612:
1577:
1517:
1370:
1350:
1323:
1112:
1092:
1072:
1049:
1029:
1004:
972:even and odd functions
963:
945:
843:
788:
704:
677:
561:
530:
523:
491:
471:
432:
345:
306:
282:
259:
143:
118:Fourier sine transform
113:
40:
5696:
5602:
5573:
5537:
5488:
5456:
5420:
5391:
5355:
5261:of its input while a
5220:
5205:
5107:
5016:
4618:
4545:
4525:
4398:
4264:
4238:
4212:
3975:
3949:
3910:
3712:
3688:
3668:
3515:
3479:
3443:
3423:
3403:
3381:
3241:
3219:
3081:
3059:
2863:
2835:
2577:
2549:
2529:
2509:
2489:
2388:
2287:
2166:
2030:
1885:
1677:
1657:
1613:
1578:
1528:
1518:
1371:
1351:
1324:
1113:
1093:
1073:
1056:is twice its integral
1050:
1030:
1005:
946:
844:
804:
789:
705:
678:
562:
524:
492:
472:
443:
433:
346:
307:
283:
260:
144:
110:
29:sine and cosine waves
22:
6005:Mathematical physics
5760:cnyack.homestead.com
5628:
5591:
5546:
5506:
5471:
5445:
5418:{\displaystyle \xi }
5409:
5364:
5324:
5277:Numerical evaluation
5120:
5029:
4628:
4561:
4534:
4418:
4273:
4247:
4221:
3984:
3958:
3925:
3721:
3701:
3677:
3544:
3488:
3452:
3432:
3421:{\displaystyle \xi }
3412:
3392:
3256:
3230:
3094:
3070:
2874:
2861:{\displaystyle \xi }
2852:
2590:
2566:
2538:
2518:
2498:
2397:
2296:
2175:
2057:
1898:
1688:
1666:
1630:
1587:
1549:
1386:
1360:
1335:
1124:
1102:
1082:
1062:
1039:
1014:
978:
854:
813:
716:
703:{\displaystyle \xi }
694:
575:
560:{\displaystyle f(t)}
542:
504:
481:
452:
446:rectangular function
357:
344:{\displaystyle \xi }
335:
305:{\displaystyle \xi }
296:
272:
157:
142:{\displaystyle f(t)}
124:
73:complex exponentials
5995:Integral transforms
4881:
4804:
4680:
4624:original function):
4470:
4305:
4262:{\displaystyle x=t}
4122:
4022:
4007:
3839:
3821:
3597:
3579:
3356: only if
3291:
3194: only if
3129:
2990:
2909:
2740:
2629:
2446:
2345:
2235:
2117:
1949:
1829:
1739:
1437:
1265:
1234:even·even=even
1175:
626:
208:
5691:
5597:
5568:
5532:
5483:
5465:radians per second
5451:
5415:
5386:
5350:
5250:negative frequency
5235:
5200:
5102:
5011:
5009:
4864:
4787:
4663:
4613:
4540:
4520:
4517:
4453:
4393:
4288:
4259:
4233:
4207:
4105:
4008:
3990:
3970:
3944:
3905:
3822:
3807:
3751:
3734:
3707:
3683:
3663:
3580:
3562:
3510:
3474:
3438:
3418:
3398:
3376:
3274:
3236:
3214:
3112:
3076:
3054:
2976:
2895:
2858:
2830:
2825:
2804:
2723:
2714:
2693:
2612:
2572:
2544:
2524:
2504:
2484:
2432:
2383:
2331:
2282:
2221:
2161:
2103:
2025:
1932:
1880:
1815:
1722:
1672:
1652:
1624:
1608:
1573:
1513:
1420:
1366:
1346:
1319:
1251:
1158:
1108:
1088:
1068:
1045:
1025:
1000:
964:
941:
899:
839:
784:
700:
673:
609:
557:
531:
519:
487:
467:
465:
428:
341:
302:
278:
255:
191:
139:
114:
81:negative frequency
75:and don't require
53:integral equations
41:
5895:978-0-521-06794-2
5797:978-1-4398-9846-8
5676:
5666:
5661:
5645:
5622:Fourier transform
5600:{\displaystyle f}
5559:
5435:angular frequency
5377:
5179:
5133:
5085:
5042:
4985:
4950:
4770:
4644:
4543:{\displaystyle i}
4434:
4412:Fourier transform
4366:
4195:
3736:
3733:
3710:{\displaystyle t}
3695:bounded variation
3686:{\displaystyle f}
3501:
3465:
3441:{\displaystyle f}
3401:{\displaystyle t}
3374:
3357:
3302:
3239:{\displaystyle f}
3212:
3195:
3140:
3079:{\displaystyle f}
3001:
2920:
2810:
2751:
2721:
2719:
2699:
2640:
2610:
2608:
2575:{\displaystyle f}
2558:Fourier inversion
2547:{\displaystyle x}
2527:{\displaystyle t}
2430:
2329:
2219:
2218:
2188:
2101:
2100:
2070:
2045:Other conventions
2011:
2009:
2008:even·odd=odd
2004:
1961:
1911:
1837:
1801:
1799:
1798:odd·odd=even
1794:
1751:
1701:
1675:{\displaystyle t}
1640:
1499:
1497:
1496:odd·even=odd
1492:
1449:
1399:
1273:
1237:
1235:
1230:
1187:
1137:
1111:{\displaystyle t}
1071:{\displaystyle 0}
988:
937:
900:
898:
867:
807:Gaussian function
760:
729:
588:
490:{\displaystyle a}
464:
407:
370:
322:spatial frequency
281:{\displaystyle t}
170:
89:signal processing
65:Fourier transform
59:representing the
6012:
6000:Fourier analysis
5980:
5973:
5967:
5966:
5964:
5963:
5957:
5946:
5937:
5931:
5930:
5928:
5927:
5905:
5899:
5898:
5865:
5859:
5858:
5850:
5844:
5843:
5830:
5824:
5823:
5817:
5809:
5781:
5775:
5774:
5772:
5771:
5762:. Archived from
5751:
5745:
5726:
5701:
5700:
5698:
5697:
5692:
5678:
5677:
5669:
5664:
5663:
5662:
5660:
5659:
5653:
5648:
5643:
5618:
5612:
5606:
5604:
5603:
5598:
5585:
5579:
5577:
5575:
5574:
5569:
5567:
5566:
5561:
5560:
5552:
5541:
5539:
5538:
5533:
5522:
5521:
5516:
5515:
5500:
5494:
5492:
5490:
5489:
5484:
5463:
5460:
5458:
5457:
5452:
5424:
5422:
5421:
5416:
5403:
5397:
5395:
5393:
5392:
5387:
5385:
5384:
5379:
5378:
5370:
5359:
5357:
5356:
5351:
5340:
5339:
5334:
5333:
5318:
5209:
5207:
5206:
5201:
5195:
5181:
5180:
5172:
5164:
5141:
5140:
5135:
5134:
5126:
5111:
5109:
5108:
5103:
5101:
5087:
5086:
5078:
5070:
5050:
5049:
5044:
5043:
5035:
5020:
5018:
5017:
5012:
5010:
4993:
4992:
4987:
4986:
4978:
4958:
4957:
4952:
4951:
4943:
4933:
4929:
4925:
4880:
4875:
4852:
4848:
4803:
4798:
4775:
4771:
4768:
4765:
4757:
4753:
4679:
4674:
4646:
4645:
4637:
4622:
4620:
4619:
4614:
4579:
4578:
4549:
4547:
4546:
4541:
4529:
4527:
4526:
4521:
4518:
4507:
4506:
4469:
4464:
4436:
4435:
4427:
4402:
4400:
4399:
4394:
4367:
4365:
4364:
4363:
4342:
4341:
4326:
4325:
4315:
4307:
4304:
4299:
4268:
4266:
4265:
4260:
4242:
4240:
4239:
4234:
4216:
4214:
4213:
4208:
4196:
4194:
4193:
4192:
4171:
4170:
4155:
4154:
4144:
4136:
4121:
4116:
4038:
4037:
4021:
4016:
4006:
4001:
3980:is fixed. Then
3979:
3977:
3976:
3971:
3953:
3951:
3950:
3945:
3943:
3942:
3914:
3912:
3911:
3906:
3838:
3833:
3820:
3815:
3800:
3796:
3750:
3735:
3726:
3717:, in which case
3716:
3714:
3713:
3708:
3692:
3690:
3689:
3684:
3672:
3670:
3669:
3664:
3596:
3591:
3578:
3573:
3519:
3517:
3516:
3511:
3509:
3508:
3503:
3502:
3494:
3483:
3481:
3480:
3475:
3473:
3472:
3467:
3466:
3458:
3447:
3445:
3444:
3439:
3427:
3425:
3424:
3419:
3407:
3405:
3404:
3399:
3385:
3383:
3382:
3377:
3375:
3372:
3358:
3355:
3310:
3309:
3304:
3303:
3295:
3290:
3285:
3245:
3243:
3242:
3237:
3223:
3221:
3220:
3215:
3213:
3210:
3196:
3193:
3148:
3147:
3142:
3141:
3133:
3128:
3123:
3085:
3083:
3082:
3077:
3063:
3061:
3060:
3055:
3009:
3008:
3003:
3002:
2994:
2989:
2984:
2928:
2927:
2922:
2921:
2913:
2908:
2903:
2867:
2865:
2864:
2859:
2839:
2837:
2836:
2831:
2824:
2811:
2808:
2805:
2800:
2759:
2758:
2753:
2752:
2744:
2739:
2734:
2713:
2700:
2697:
2694:
2689:
2648:
2647:
2642:
2641:
2633:
2628:
2623:
2581:
2579:
2578:
2573:
2553:
2551:
2550:
2545:
2533:
2531:
2530:
2525:
2513:
2511:
2510:
2505:
2493:
2491:
2490:
2485:
2445:
2440:
2431:
2423:
2409:
2408:
2392:
2390:
2389:
2384:
2344:
2339:
2330:
2322:
2308:
2307:
2291:
2289:
2288:
2283:
2234:
2229:
2220:
2211:
2210:
2196:
2195:
2190:
2189:
2181:
2170:
2168:
2167:
2162:
2116:
2111:
2102:
2093:
2092:
2078:
2077:
2072:
2071:
2063:
2034:
2032:
2031:
2026:
2010:
2007:
2005:
2000:
1963:
1962:
1959:
1952:
1950:
1948:
1943:
1919:
1918:
1913:
1912:
1904:
1889:
1887:
1886:
1881:
1839:
1838:
1835:
1828:
1823:
1800:
1797:
1795:
1790:
1753:
1752:
1749:
1742:
1740:
1738:
1733:
1709:
1708:
1703:
1702:
1694:
1681:
1679:
1678:
1673:
1661:
1659:
1658:
1653:
1642:
1641:
1638:
1617:
1615:
1614:
1609:
1582:
1580:
1579:
1574:
1522:
1520:
1519:
1514:
1498:
1495:
1493:
1488:
1451:
1450:
1447:
1440:
1438:
1436:
1431:
1407:
1406:
1401:
1400:
1392:
1375:
1373:
1372:
1367:
1355:
1353:
1352:
1347:
1342:
1328:
1326:
1325:
1320:
1275:
1274:
1271:
1264:
1259:
1236:
1233:
1231:
1226:
1189:
1188:
1185:
1178:
1176:
1174:
1169:
1145:
1144:
1139:
1138:
1130:
1117:
1115:
1114:
1109:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1054:
1052:
1051:
1046:
1034:
1032:
1031:
1026:
1021:
1009:
1007:
1006:
1001:
990:
989:
986:
961:
950:
948:
947:
942:
940:
939:
938:
933:
932:
931:
912:
901:
891:
889:
875:
874:
869:
868:
860:
848:
846:
845:
840:
838:
837:
836:
835:
793:
791:
790:
785:
768:
767:
762:
761:
753:
737:
736:
731:
730:
722:
709:
707:
706:
701:
682:
680:
679:
674:
625:
620:
596:
595:
590:
589:
581:
566:
564:
563:
558:
528:
526:
525:
520:
496:
494:
493:
488:
476:
474:
473:
468:
466:
457:
437:
435:
434:
429:
415:
414:
409:
408:
400:
378:
377:
372:
371:
363:
350:
348:
347:
342:
311:
309:
308:
303:
287:
285:
284:
279:
264:
262:
261:
256:
207:
202:
178:
177:
172:
171:
163:
148:
146:
145:
140:
97:Fourier analysis
25:frequency domain
6020:
6019:
6015:
6014:
6013:
6011:
6010:
6009:
5985:
5984:
5983:
5974:
5970:
5961:
5959:
5955:
5944:
5939:
5938:
5934:
5925:
5923:
5907:
5906:
5902:
5887:
5867:
5866:
5862:
5852:
5851:
5847:
5834:Poincaré, Henri
5832:
5831:
5827:
5810:
5798:
5783:
5782:
5778:
5769:
5767:
5753:
5752:
5748:
5727:
5723:
5710:
5705:
5704:
5626:
5625:
5619:
5615:
5589:
5588:
5586:
5582:
5549:
5544:
5543:
5509:
5504:
5503:
5501:
5497:
5469:
5468:
5461:
5443:
5442:
5407:
5406:
5404:
5400:
5367:
5362:
5361:
5327:
5322:
5321:
5319:
5315:
5310:
5288:
5279:
5215:
5123:
5118:
5117:
5032:
5027:
5026:
5008:
5007:
4975:
4940:
4931:
4930:
4863:
4859:
4786:
4782:
4773:
4772:
4769:Euler's Formula
4764:
4697:
4693:
4656:
4626:
4625:
4567:
4559:
4558:
4556:Euler's formula
4532:
4531:
4516:
4515:
4483:
4446:
4416:
4415:
4408:
4355:
4333:
4317:
4316:
4308:
4271:
4270:
4245:
4244:
4219:
4218:
4184:
4162:
4146:
4145:
4137:
4023:
3982:
3981:
3956:
3955:
3928:
3923:
3922:
3921:is to insert a
3756:
3752:
3719:
3718:
3699:
3698:
3675:
3674:
3542:
3541:
3530:
3522:transform pairs
3491:
3486:
3485:
3455:
3450:
3449:
3430:
3429:
3410:
3409:
3390:
3389:
3292:
3254:
3253:
3228:
3227:
3130:
3092:
3091:
3068:
3067:
2991:
2910:
2872:
2871:
2850:
2849:
2846:
2844:Simplifications
2841:
2741:
2722:
2630:
2611:
2588:
2587:
2564:
2563:
2560:
2536:
2535:
2516:
2515:
2496:
2495:
2400:
2395:
2394:
2299:
2294:
2293:
2178:
2173:
2172:
2060:
2055:
2054:
2047:
1954:
1953:
1901:
1896:
1895:
1830:
1744:
1743:
1691:
1686:
1685:
1664:
1663:
1633:
1628:
1627:
1585:
1584:
1547:
1546:
1442:
1441:
1389:
1384:
1383:
1358:
1357:
1333:
1332:
1266:
1180:
1179:
1127:
1122:
1121:
1100:
1099:
1080:
1079:
1060:
1059:
1037:
1036:
1012:
1011:
981:
976:
975:
953:
951:
923:
913:
903:
857:
852:
851:
850:
827:
816:
811:
810:
799:
750:
719:
714:
713:
692:
691:
684:
578:
573:
572:
540:
539:
533:
502:
501:
499:normalized sinc
479:
478:
450:
449:
397:
360:
355:
354:
333:
332:
294:
293:
270:
269:
266:
160:
155:
154:
122:
121:
105:
77:complex numbers
17:
12:
11:
5:
6018:
6016:
6008:
6007:
6002:
5997:
5987:
5986:
5982:
5981:
5968:
5932:
5900:
5885:
5875:(1927-01-02).
5860:
5845:
5825:
5796:
5776:
5746:
5720:
5719:
5718:
5709:
5706:
5703:
5702:
5690:
5687:
5684:
5681:
5675:
5672:
5658:
5652:
5642:
5639:
5636:
5633:
5613:
5596:
5580:
5565:
5558:
5555:
5531:
5528:
5525:
5520:
5514:
5495:
5482:
5479:
5476:
5450:
5414:
5398:
5383:
5376:
5373:
5349:
5346:
5343:
5338:
5332:
5312:
5311:
5309:
5306:
5305:
5304:
5299:
5294:
5287:
5284:
5278:
5275:
5214:
5211:
5199:
5194:
5190:
5187:
5184:
5178:
5175:
5168:
5163:
5160:
5156:
5153:
5150:
5147:
5144:
5139:
5132:
5129:
5100:
5096:
5093:
5090:
5084:
5081:
5074:
5069:
5066:
5062:
5059:
5056:
5053:
5048:
5041:
5038:
5006:
5002:
4999:
4996:
4991:
4984:
4981:
4973:
4970:
4967:
4964:
4961:
4956:
4949:
4946:
4939:
4936:
4934:
4932:
4928:
4924:
4921:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4879:
4874:
4871:
4867:
4862:
4858:
4855:
4851:
4847:
4844:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4802:
4797:
4794:
4790:
4785:
4781:
4778:
4776:
4774:
4766:
4763:
4760:
4756:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4696:
4692:
4689:
4686:
4683:
4678:
4673:
4670:
4666:
4662:
4659:
4657:
4655:
4652:
4649:
4643:
4640:
4634:
4633:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4577:
4574:
4570:
4566:
4554:. By applying
4539:
4514:
4511:
4505:
4502:
4499:
4496:
4493:
4490:
4486:
4482:
4479:
4476:
4473:
4468:
4463:
4460:
4456:
4452:
4449:
4447:
4445:
4442:
4439:
4433:
4430:
4424:
4423:
4407:
4404:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4362:
4358:
4354:
4351:
4348:
4345:
4340:
4336:
4332:
4329:
4324:
4320:
4314:
4311:
4303:
4298:
4295:
4291:
4287:
4284:
4281:
4278:
4258:
4255:
4252:
4232:
4229:
4226:
4206:
4203:
4200:
4191:
4187:
4183:
4180:
4177:
4174:
4169:
4165:
4161:
4158:
4153:
4149:
4143:
4140:
4134:
4131:
4128:
4125:
4120:
4115:
4112:
4108:
4104:
4101:
4098:
4094:
4091:
4088:
4085:
4081:
4078:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4036:
4033:
4030:
4026:
4020:
4015:
4011:
4005:
4000:
3997:
3993:
3989:
3969:
3966:
3963:
3941:
3938:
3935:
3931:
3904:
3901:
3898:
3894:
3891:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3837:
3832:
3829:
3825:
3819:
3814:
3810:
3806:
3803:
3799:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3755:
3749:
3746:
3743:
3739:
3732:
3729:
3706:
3682:
3662:
3659:
3656:
3652:
3649:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3595:
3590:
3587:
3583:
3577:
3572:
3569:
3565:
3561:
3558:
3555:
3552:
3549:
3529:
3526:
3507:
3500:
3497:
3471:
3464:
3461:
3437:
3417:
3397:
3373: is even.
3370:
3367:
3364:
3361:
3353:
3350:
3347:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3308:
3301:
3298:
3289:
3284:
3281:
3277:
3273:
3270:
3267:
3264:
3261:
3235:
3208:
3205:
3202:
3199:
3191:
3188:
3185:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3146:
3139:
3136:
3127:
3122:
3119:
3115:
3111:
3108:
3105:
3102:
3099:
3075:
3053:
3049:
3046:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3007:
3000:
2997:
2988:
2983:
2979:
2975:
2972:
2968:
2965:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2919:
2916:
2907:
2902:
2898:
2894:
2891:
2888:
2885:
2882:
2879:
2857:
2845:
2842:
2829:
2823:
2820:
2817:
2814:
2803:
2799:
2796:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2757:
2750:
2747:
2738:
2733:
2730:
2726:
2718:
2712:
2709:
2706:
2703:
2692:
2688:
2685:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2646:
2639:
2636:
2627:
2622:
2619:
2615:
2607:
2604:
2601:
2598:
2595:
2584:
2571:
2559:
2556:
2543:
2523:
2503:
2483:
2480:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2449:
2444:
2439:
2435:
2429:
2426:
2421:
2418:
2415:
2412:
2407:
2403:
2382:
2379:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2343:
2338:
2334:
2328:
2325:
2320:
2317:
2314:
2311:
2306:
2302:
2281:
2278:
2275:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2233:
2228:
2224:
2217:
2214:
2208:
2205:
2202:
2199:
2194:
2187:
2184:
2160:
2157:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2115:
2110:
2106:
2099:
2096:
2090:
2087:
2084:
2081:
2076:
2069:
2066:
2046:
2043:
2024:
2021:
2018:
2015:
2003:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1957:
1947:
1942:
1939:
1935:
1931:
1928:
1925:
1922:
1917:
1910:
1907:
1879:
1876:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1833:
1827:
1822:
1818:
1814:
1811:
1808:
1805:
1793:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1747:
1737:
1732:
1729:
1725:
1721:
1718:
1715:
1712:
1707:
1700:
1697:
1671:
1651:
1648:
1645:
1636:
1620:transform pair
1607:
1604:
1601:
1598:
1595:
1592:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1512:
1509:
1506:
1503:
1491:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1445:
1435:
1430:
1427:
1423:
1419:
1416:
1413:
1410:
1405:
1398:
1395:
1365:
1345:
1341:
1318:
1315:
1312:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1269:
1263:
1258:
1254:
1250:
1247:
1244:
1241:
1229:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1183:
1173:
1168:
1165:
1161:
1157:
1154:
1151:
1148:
1143:
1136:
1133:
1107:
1087:
1067:
1044:
1024:
1020:
999:
996:
993:
984:
936:
930:
926:
922:
919:
916:
910:
906:
897:
894:
887:
884:
881:
878:
873:
866:
863:
834:
830:
826:
823:
819:
798:
795:
783:
780:
777:
774:
771:
766:
759:
756:
749:
746:
743:
740:
735:
728:
725:
699:
672:
669:
666:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
624:
619:
616:
612:
608:
605:
602:
599:
594:
587:
584:
569:
556:
553:
550:
547:
529:plotted above.
518:
515:
512:
509:
486:
463:
460:
427:
424:
421:
418:
413:
406:
403:
396:
393:
390:
387:
384:
381:
376:
369:
366:
340:
301:
277:
254:
251:
248:
244:
241:
238:
235:
232:
229:
226:
223:
220:
217:
214:
211:
206:
201:
198:
194:
190:
187:
184:
181:
176:
169:
166:
151:
138:
135:
132:
129:
104:
101:
85:Joseph Fourier
47:, the Fourier
15:
13:
10:
9:
6:
4:
3:
2:
6017:
6006:
6003:
6001:
5998:
5996:
5993:
5992:
5990:
5978:
5972:
5969:
5954:
5950:
5943:
5936:
5933:
5921:
5917:
5916:
5911:
5904:
5901:
5896:
5892:
5888:
5886:0-521-06794-4
5882:
5878:
5874:
5870:
5864:
5861:
5856:
5849:
5846:
5841:
5840:
5835:
5829:
5826:
5821:
5815:
5807:
5803:
5799:
5793:
5789:
5788:
5780:
5777:
5766:on 2023-06-07
5765:
5761:
5757:
5750:
5747:
5744:
5743:0-471-04409-1
5740:
5736:
5735:
5730:
5725:
5722:
5716:
5712:
5711:
5707:
5688:
5682:
5673:
5670:
5637:
5631:
5623:
5617:
5614:
5610:
5594:
5584:
5581:
5563:
5553:
5526:
5518:
5499:
5496:
5480:
5477:
5474:
5466:
5448:
5440:
5436:
5432:
5428:
5412:
5402:
5399:
5381:
5371:
5344:
5336:
5317:
5314:
5307:
5303:
5300:
5298:
5295:
5293:
5290:
5289:
5285:
5283:
5276:
5274:
5272:
5268:
5265:may start by
5264:
5260:
5257:may start by
5256:
5251:
5246:
5244:
5240:
5232:
5228:
5224:
5223:phase-shifted
5219:
5213:Pros and cons
5212:
5210:
5197:
5185:
5173:
5154:
5151:
5145:
5137:
5127:
5115:
5091:
5079:
5060:
5054:
5046:
5036:
5024:
5004:
4997:
4989:
4979:
4971:
4968:
4962:
4954:
4944:
4937:
4935:
4926:
4922:
4919:
4912:
4909:
4906:
4903:
4897:
4894:
4888:
4882:
4869:
4865:
4860:
4856:
4853:
4849:
4845:
4842:
4835:
4832:
4829:
4826:
4820:
4817:
4811:
4805:
4792:
4788:
4783:
4779:
4777:
4761:
4758:
4754:
4747:
4744:
4741:
4738:
4732:
4729:
4725:
4722:
4716:
4713:
4710:
4707:
4701:
4698:
4694:
4687:
4681:
4668:
4664:
4660:
4658:
4650:
4638:
4610:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4575:
4572:
4568:
4557:
4553:
4537:
4512:
4509:
4503:
4500:
4497:
4494:
4491:
4488:
4484:
4477:
4471:
4458:
4454:
4450:
4448:
4440:
4428:
4413:
4405:
4403:
4390:
4384:
4378:
4375:
4372:
4369:
4360:
4352:
4349:
4346:
4338:
4334:
4330:
4327:
4322:
4318:
4312:
4309:
4293:
4289:
4282:
4276:
4256:
4253:
4250:
4230:
4224:
4204:
4201:
4198:
4189:
4181:
4178:
4175:
4167:
4163:
4159:
4156:
4151:
4147:
4141:
4138:
4129:
4123:
4110:
4106:
4102:
4099:
4096:
4089:
4083:
4079:
4076:
4066:
4063:
4060:
4054:
4051:
4048:
4042:
4039:
4034:
4031:
4028:
4024:
4013:
4009:
3995:
3991:
3987:
3967:
3964:
3961:
3939:
3936:
3933:
3929:
3920:
3915:
3902:
3899:
3896:
3892:
3889:
3879:
3876:
3873:
3867:
3864:
3861:
3855:
3852:
3846:
3840:
3827:
3823:
3812:
3808:
3804:
3801:
3797:
3790:
3787:
3784:
3778:
3775:
3769:
3766:
3763:
3757:
3753:
3747:
3741:
3730:
3727:
3704:
3696:
3680:
3660:
3657:
3654:
3650:
3647:
3637:
3634:
3631:
3625:
3622:
3619:
3613:
3610:
3604:
3598:
3585:
3581:
3567:
3563:
3559:
3553:
3547:
3539:
3535:
3527:
3525:
3523:
3505:
3495:
3469:
3459:
3448:swapped with
3435:
3415:
3408:swapped with
3395:
3386:
3365:
3359:
3351:
3348:
3345:
3338:
3335:
3332:
3329:
3323:
3320:
3314:
3306:
3296:
3279:
3275:
3271:
3265:
3259:
3251:
3249:
3248:even function
3233:
3224:
3211: is odd.
3203:
3197:
3189:
3186:
3183:
3176:
3173:
3170:
3167:
3161:
3158:
3152:
3144:
3134:
3117:
3113:
3109:
3103:
3097:
3089:
3073:
3064:
3051:
3047:
3044:
3037:
3034:
3031:
3028:
3022:
3019:
3013:
3005:
2995:
2981:
2977:
2973:
2970:
2966:
2963:
2956:
2953:
2950:
2947:
2941:
2938:
2932:
2924:
2914:
2900:
2896:
2892:
2889:
2883:
2877:
2869:
2855:
2843:
2840:
2827:
2818:
2812:
2801:
2797:
2794:
2787:
2784:
2781:
2778:
2772:
2769:
2763:
2755:
2745:
2728:
2724:
2716:
2707:
2701:
2690:
2686:
2683:
2676:
2673:
2670:
2667:
2661:
2658:
2652:
2644:
2634:
2617:
2613:
2605:
2599:
2593:
2583:
2569:
2557:
2555:
2541:
2521:
2501:
2481:
2478:
2471:
2468:
2462:
2459:
2453:
2447:
2437:
2433:
2427:
2424:
2419:
2413:
2405:
2401:
2380:
2377:
2370:
2367:
2361:
2358:
2352:
2346:
2336:
2332:
2326:
2323:
2318:
2312:
2304:
2300:
2279:
2276:
2273:
2266:
2263:
2260:
2257:
2251:
2248:
2242:
2236:
2226:
2222:
2215:
2212:
2206:
2200:
2192:
2182:
2158:
2155:
2148:
2145:
2142:
2139:
2133:
2130:
2124:
2118:
2108:
2104:
2097:
2094:
2088:
2082:
2074:
2064:
2052:
2044:
2042:
2040:
2035:
2022:
2019:
2016:
2013:
2001:
1994:
1991:
1988:
1985:
1979:
1976:
1973:
1967:
1955:
1937:
1933:
1929:
1923:
1915:
1905:
1893:
1890:
1877:
1874:
1867:
1864:
1861:
1858:
1852:
1849:
1843:
1831:
1820:
1816:
1812:
1809:
1806:
1803:
1791:
1784:
1781:
1778:
1775:
1769:
1766:
1763:
1757:
1745:
1727:
1723:
1719:
1713:
1705:
1695:
1683:
1669:
1646:
1634:
1621:
1602:
1599:
1593:
1590:
1570:
1564:
1561:
1555:
1552:
1544:
1540:
1536:
1532:
1527:
1523:
1510:
1507:
1504:
1501:
1489:
1482:
1479:
1476:
1473:
1467:
1464:
1461:
1455:
1443:
1425:
1421:
1417:
1411:
1403:
1393:
1381:
1379:
1339:
1329:
1316:
1313:
1310:
1303:
1300:
1297:
1294:
1288:
1285:
1279:
1267:
1256:
1252:
1248:
1245:
1242:
1239:
1227:
1220:
1217:
1214:
1211:
1205:
1202:
1199:
1193:
1181:
1163:
1159:
1155:
1149:
1141:
1131:
1119:
1105:
1065:
1057:
1018:
994:
982:
973:
969:
960:
956:
934:
928:
920:
917:
908:
904:
895:
892:
885:
879:
871:
861:
832:
828:
824:
821:
817:
808:
803:
796:
794:
781:
775:
772:
764:
754:
747:
741:
733:
723:
711:
697:
689:
688:even function
683:
670:
667:
664:
657:
654:
651:
648:
642:
639:
633:
627:
614:
610:
606:
600:
592:
582:
568:
551:
545:
537:
513:
510:
500:
484:
461:
458:
447:
442:
438:
425:
419:
411:
401:
394:
391:
385:
382:
374:
364:
352:
338:
330:
325:
323:
319:
315:
299:
291:
275:
265:
252:
249:
246:
239:
236:
233:
230:
224:
221:
215:
209:
196:
192:
188:
182:
174:
164:
150:
133:
127:
119:
112:respectively.
109:
102:
100:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
61:odd component
58:
54:
50:
46:
38:
34:
30:
26:
21:
5976:
5971:
5960:. Retrieved
5948:
5935:
5924:. Retrieved
5913:
5903:
5876:
5863:
5854:
5848:
5838:
5828:
5786:
5779:
5768:. Retrieved
5764:the original
5759:
5749:
5732:
5729:Mary L. Boas
5724:
5714:
5616:
5583:
5498:
5401:
5316:
5280:
5247:
5242:
5236:
5230:
5113:
4409:
3916:
3537:
3531:
3521:
3387:
3252:
3225:
3088:odd function
3065:
2870:
2847:
2586:
2561:
2048:
2036:
1894:
1891:
1684:
1625:
1619:
1382:
1330:
1120:
965:
958:
954:
712:
685:
571:
535:
532:
353:
329:odd function
326:
267:
153:
117:
115:
48:
42:
5542:instead of
5360:instead of
448:(of height
45:mathematics
5989:Categories
5962:2024-09-11
5926:2024-09-09
5915:IEEE Pulse
5770:2018-10-08
5708:References
3428:(and with
1537:about the
477:and width
103:Definition
93:statistics
67:concisely
57:sine waves
5814:cite book
5806:822959644
5683:ξ
5674:^
5651:⟷
5557:^
5481:ξ
5478:π
5449:ω
5413:ξ
5375:^
5186:ξ
5177:^
5155:−
5146:ξ
5131:^
5092:ξ
5083:^
5055:ξ
5040:^
4998:ξ
4983:^
4969:−
4963:ξ
4948:^
4910:ξ
4907:π
4898:
4878:∞
4873:∞
4870:−
4866:∫
4854:−
4833:ξ
4830:π
4821:
4801:∞
4796:∞
4793:−
4789:∫
4745:ξ
4742:π
4733:
4723:−
4714:ξ
4711:π
4702:
4677:∞
4672:∞
4669:−
4665:∫
4651:ξ
4642:^
4602:
4587:
4501:ξ
4495:π
4489:−
4467:∞
4462:∞
4459:−
4455:∫
4441:ξ
4432:^
4350:−
4335:π
4319:δ
4313:δ
4302:∞
4297:∞
4294:−
4290:∫
4228:→
4225:δ
4217:Now when
4179:−
4164:π
4148:δ
4142:δ
4119:∞
4114:∞
4111:−
4107:∫
4080:ξ
4064:−
4055:ξ
4052:π
4043:
4035:ξ
4032:δ
4029:−
4019:∞
4010:∫
4004:∞
3999:∞
3996:−
3992:∫
3962:δ
3940:ξ
3937:δ
3934:−
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3877:−
3868:ξ
3865:π
3856:
3836:∞
3831:∞
3828:−
3824:∫
3818:∞
3809:∫
3788:−
3745:→
3658:ξ
3635:−
3626:ξ
3623:π
3614:
3594:∞
3589:∞
3586:−
3582:∫
3576:∞
3571:∞
3568:−
3564:∫
3499:^
3463:^
3416:ξ
3349:ξ
3336:ξ
3333:π
3324:
3315:ξ
3300:^
3288:∞
3283:∞
3280:−
3276:∫
3187:ξ
3174:ξ
3171:π
3162:
3153:ξ
3138:^
3126:∞
3121:∞
3118:−
3114:∫
3066:Also, if
3048:ξ
3035:ξ
3032:π
3023:
3014:ξ
2999:^
2987:∞
2978:∫
2967:ξ
2954:ξ
2951:π
2942:
2933:ξ
2918:^
2906:∞
2897:∫
2856:ξ
2802:⏟
2798:ξ
2785:ξ
2782:π
2773:
2764:ξ
2749:^
2737:∞
2732:∞
2729:−
2725:∫
2691:⏟
2687:ξ
2674:ξ
2671:π
2662:
2653:ξ
2638:^
2626:∞
2621:∞
2618:−
2614:∫
2502:α
2469:α
2463:
2443:∞
2434:∫
2428:π
2414:α
2368:α
2362:
2342:∞
2333:∫
2327:π
2313:α
2264:ξ
2261:π
2252:
2232:∞
2223:∫
2216:π
2201:ξ
2186:^
2146:ξ
2143:π
2134:
2114:∞
2105:∫
2098:π
2083:ξ
2068:^
2002:⏞
1992:ξ
1989:π
1980:
1974:⋅
1946:∞
1941:∞
1938:−
1934:∫
1924:ξ
1909:^
1865:ξ
1862:π
1853:
1826:∞
1817:∫
1792:⏞
1782:ξ
1779:π
1770:
1764:⋅
1736:∞
1731:∞
1728:−
1724:∫
1714:ξ
1699:^
1603:ξ
1594:
1565:ξ
1556:
1490:⏞
1480:ξ
1477:π
1468:
1462:⋅
1434:∞
1429:∞
1426:−
1422:∫
1412:ξ
1397:^
1364:∞
1344:∞
1340:−
1301:ξ
1298:π
1289:
1262:∞
1253:∫
1228:⏞
1218:ξ
1215:π
1206:
1200:⋅
1172:∞
1167:∞
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1160:∫
1150:ξ
1135:^
1086:∞
1043:∞
1023:∞
1019:−
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921:ξ
918:π
909:−
896:α
893:π
880:ξ
865:^
825:α
822:−
776:ξ
773:−
758:^
742:ξ
727:^
698:ξ
655:ξ
652:π
643:
623:∞
618:∞
615:−
611:∫
601:ξ
586:^
514:ξ
497:) is the
420:ξ
405:^
395:−
386:ξ
383:−
368:^
339:ξ
314:frequency
300:ξ
237:ξ
234:π
225:
205:∞
200:∞
197:−
193:∫
183:ξ
168:^
5953:Archived
5920:Archived
5836:(1895).
5429:and the
5286:See also
5231:together
5114:negative
318:position
69:contains
5467:equals
5439:radians
4550:is the
1535:degrees
1531:rotated
292:, then
39:domain.
37:spatial
5893:
5883:
5804:
5794:
5741:
5665:
5644:
5462:
5431:second
4530:where
3919:Cauchy
3534:cosine
3246:is an
3086:is an
2494:using
1539:origin
288:means
5956:(PDF)
5945:(PDF)
5427:Hertz
5308:Notes
5239:phase
1058:from
5891:ISBN
5881:ISBN
5820:link
5802:OCLC
5792:ISBN
5739:ISBN
3965:>
1960:even
1533:180
1272:even
1186:even
987:even
966:The
567:is:
534:The
320:and
290:time
149:is:
116:The
91:and
51:are
33:time
5243:and
4895:sin
4818:cos
4730:sin
4699:cos
4599:sin
4584:cos
4040:cos
3853:cos
3738:lim
3611:cos
3484:or
3321:cos
3159:sin
3020:cos
2939:sin
2770:cos
2659:sin
2460:sin
2359:cos
2249:sin
2131:cos
1977:sin
1850:sin
1836:odd
1767:sin
1750:odd
1639:odd
1591:sin
1553:sin
1465:cos
1448:odd
1376:of
1356:to
1286:cos
1203:cos
1078:to
1035:to
640:cos
538:of
324:).
312:is
268:If
222:sin
120:of
99:.
79:or
43:In
35:or
5991::
5951:.
5947:.
5918:.
5912:.
5889:.
5871:;
5816:}}
5812:{{
5800:.
5758:.
5731:,
5273:.
3540::
3524:.
2023:0.
1682::
1511:0.
1118::
710::
351::
5965:.
5929:.
5897:.
5822:)
5808:.
5773:.
5689:.
5686:)
5680:(
5671:f
5657:F
5641:)
5638:t
5635:(
5632:f
5611:.
5595:f
5578:.
5564:c
5554:f
5530:)
5527:f
5524:(
5519:c
5513:F
5493:.
5475:2
5396:.
5382:s
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5342:(
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5198:.
5193:]
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5167:[
5162:m
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5149:)
5143:(
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5099:]
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5089:(
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5061:=
5058:)
5052:(
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5005:.
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4938:=
4927:)
4923:t
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4913:t
4904:2
4901:(
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4889:t
4886:(
4883:f
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4751:)
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4736:(
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4717:t
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4581:=
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4280:(
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4257:t
4254:=
4251:x
4231:0
4205:.
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4103:=
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3802:=
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3560:=
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3318:)
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3272:=
3269:)
3266:t
3263:(
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3135:f
3110:=
3107:)
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3101:(
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3052:.
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3041:)
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3026:(
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2996:f
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2776:(
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2705:(
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2684:d
2680:)
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2668:2
2665:(
2656:)
2650:(
2645:s
2635:f
2606:=
2603:)
2600:t
2597:(
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2570:f
2542:x
2522:t
2482:x
2479:d
2475:)
2472:x
2466:(
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2451:(
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2420:=
2417:)
2411:(
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2371:x
2365:(
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2350:(
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2337:0
2324:2
2319:=
2316:)
2310:(
2305:c
2301:F
2280:.
2277:t
2274:d
2270:)
2267:t
2258:2
2255:(
2246:)
2243:t
2240:(
2237:f
2227:0
2213:2
2207:=
2204:)
2198:(
2193:s
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2159:t
2156:d
2152:)
2149:t
2140:2
2137:(
2128:)
2125:t
2122:(
2119:f
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2095:2
2089:=
2086:)
2080:(
2075:c
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2020:=
2017:t
2014:d
1998:)
1995:t
1986:2
1983:(
1971:)
1968:t
1965:(
1956:f
1930:=
1927:)
1921:(
1916:s
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1878:t
1875:d
1871:)
1868:t
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1856:(
1847:)
1844:t
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1832:f
1821:0
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1810:=
1807:t
1804:d
1788:)
1785:t
1776:2
1773:(
1761:)
1758:t
1755:(
1746:f
1720:=
1717:)
1711:(
1706:s
1696:f
1670:t
1650:)
1647:t
1644:(
1635:f
1622:.
1606:)
1600:a
1597:(
1571:.
1568:)
1562:a
1559:(
1508:=
1505:t
1502:d
1486:)
1483:t
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1471:(
1459:)
1456:t
1453:(
1444:f
1418:=
1415:)
1409:(
1404:c
1394:f
1317:.
1314:t
1311:d
1307:)
1304:t
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1292:(
1283:)
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1268:f
1257:0
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1246:=
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1224:)
1221:t
1212:2
1209:(
1197:)
1194:t
1191:(
1182:f
1156:=
1153:)
1147:(
1142:c
1132:f
1106:t
1066:0
998:)
995:t
992:(
983:f
959:π
957:=
955:α
929:2
925:)
915:(
905:e
886:=
883:)
877:(
872:c
862:f
833:2
829:t
818:e
782:.
779:)
770:(
765:c
755:f
748:=
745:)
739:(
734:c
724:f
671:.
668:t
665:d
661:)
658:t
649:2
646:(
637:)
634:t
631:(
628:f
607:=
604:)
598:(
593:c
583:f
555:)
552:t
549:(
546:f
517:)
511:a
508:(
485:a
462:a
459:1
426:.
423:)
417:(
412:s
402:f
392:=
389:)
380:(
375:s
365:f
276:t
253:.
250:t
247:d
243:)
240:t
231:2
228:(
219:)
216:t
213:(
210:f
189:=
186:)
180:(
175:s
165:f
137:)
134:t
131:(
128:f
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