84:
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53:
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22:
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3823:
than point to the article you mentioned. And there isn't an immediate analogue with the Dirac delta function as that can be treated as equivalent to a probability distribution in it own right, whereas a Dirac comb cannot ... because it integrates to infinity and so is not equivalent to a probability distribution.
3822:
It would help if the article gave a proper indication of the supposed relationship to
Directional statistics.... there was nothing at all there before this latest edit. It would be good at least to have a section headed "use in directional statistics", or some such thing, even if it does little more
4138:
one presumes is the Dirac delta function, but this is not a page in the middle of text on signal processing, rather a standalone page that should, in manner of such pages introduce terms as needed, not assume understanding. Of course I know the challenge is always where to draw the line and some
443:
Is the "scaling property" mentioned somewhere in the literature? I don't see the point of mentioning it here or how it can be of immediate used, even if it appears to be correct. In fact, they way it is presented now does not even mention the Dirac comb function explicitly. A more interesting
2675:
1239:
3198:
3418:
450:
2995:
3649:
Should the article say something about what
Bracewell calls the "bed of nails" function? This and its natural higher-dimensional analogs are very important in crystallography. It's surprising that, as far as I can tell, Knowledge lacks an article on them.
3850:
Directional (or wrapped, or circular) probability distributions are never integrated over the real number line (from negative infinity to positive infinity), they are always integrated over the unit circle (an interval of length 2π), as explained in the
3807:. The links in this article to circular statistics articles are constantly being removed because the relationship is not clear to some editors from reading this article. I have added some information in the intro, but any other ideas would be welcome.
3855:
article. Since all wrapped distributions are periodic with period 2π the limits of integration are irrelevant as long as the length of the interval is 2π. As a result, the Dirac comb of course integrates to unity. In linear statistics, the variable
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713:{\displaystyle \Delta _{T}(t)=\sum _{k=-\infty }^{\infty }\delta (t-kT)=\sum _{k=-\infty }^{\infty }\delta (T)={\frac {1}{|T|}}\sum _{k=-\infty }^{\infty }\delta (t/T-k)={\frac {1}{|T|}}\Delta _{1}(t/T)}
3278:
980:
OK, I agree that the shah symbol may appear somewhat shaky through the pseudo-latex renderer, even though it may be possible to make the wiki-guys to implement a proper shah symbol. Why not stick to
2872:
4239:
1680:
140:
3860:
often extends from -infinity to infinity. In directional statistics, the angle is in some interval of length 2π In other words the angles θ and θ+2π are considered identical. The analog of
1463:
3898:
with one of many edits by an editor who often thinks he's making the article better and isn't. Undoing his "improvements" over the project's scope would be a full-time job. The equation
3777:
This seems to beg the question a bit, depending on what one means by the composition of δ with a smooth function. This identity is a trivial consequence of the usual definition (see
1062:
2865:
1980:
928:
and mainly call it the Dirac comb function? In principle, it doesn't matter much to me which notation and names are used as long as they can be motivated. If we are going to use
322:
3615:
400:
2095:
obviously it is a periodic function with period 1/T (as it is a sum of evenly spaced delta functions, each 1/T apart in f-space). Next take the
Fourier series of x(f) by the
2724:
3902:
make sense, but as it was, is just incorrect. One quick look at the period of the the periodic function on the two sides of the equation would immediately reveal the error.
3841:
First there was the probability nav box, which was removed. So I put back the nav box along with a short explanation. I will expand the explanation, and include the nav box.
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I am puzzled by the absence of any mention that the
Fourier series of the Dirac comb can be expressed purely as a sum of cosine terms. This belongs in the article.
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to denote the Dirac comb function. If
Bracewell is the base reference, why not stick to that symbol and why not call it the shah function, instead of using
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3781:) although it is possible that there are other ways of defining the composition in which the statement becomes more interesting (I don't know).
3511:
I changed the variables in this section, the left hand sides were given as a function of t and the right hand sides were given as a function of
2670:{\displaystyle c_{n}={\frac {1}{T}}\int _{f_{0}}^{f_{0}+1/T}\left(\sum _{k=-\infty }^{\infty }\delta (f-{\frac {k}{T}})\right)e^{-i2\pi nfT}df}
2013:
1234:{\displaystyle \quad {1 \over T}\sum _{k=-\infty }^{\infty }\delta \left(f-{k \over T}\right)\quad =\sum _{n=-\infty }^{\infty }e^{-i2\pi fnT}}
211:
106:
1306:
3803:
Can someone help here - The Dirac comb of period 2π is to circular statistics what the Dirac delta function is to linear statistics. E.g see
859:
Yes, I see it, but can you please tell me what information the "Scaling property" section of the article provides about the comb function? --
3903:
3193:{\displaystyle 1=\int _{-{\frac {1}{2T}}}^{\frac {1}{2T}}\delta (f)e^{-i2\pi nfT}df=\int _{-\infty }^{\infty }\delta (f)e^{-i2\pi nfT}df,}
1829:
2104:
3944:
4243:
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97:
58:
3413:{\displaystyle \sum _{k=-\infty }^{\infty }F\left(\nu +{\frac {k}{T}}\right)=T\sum _{n=-\infty }^{\infty }f(nT)\ e^{-i2\pi nT\nu }}
2990:{\displaystyle \sum _{k=-\infty }^{\infty }\delta (f-{\frac {k}{T}})={\frac {1}{T}}\sum _{n=-\infty }^{\infty }e^{i2\pi nfT}}
3788:
3655:
958:
Bracewell isn't the only reference about the Dirac comb. Oppenheim and
Schafer is an authorative reference and they call it
781:. This property is useful since it can simplify certain derivations, and we only need to remember the Fourier transform of
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33:
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21:
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2005:
I found this very useful, but personally took me a little time to digest so want to clarify a little more.
1991:
1023:
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2223:
determined by the standard
Fourier trick (of multiplying and then integrating orthogonal functions since
3002:
1934:
1083:
I've replaced "III" with "\text{Ш}", which produces a slightly-smaller-size letter but otherwise works.
39:
3629:
286:
83:
3585:
948:
and Dirac comb function, I would like to see a reference which establishes this notation and name. --
327:
is the Shah function or whether it needs T=1. Anyway, I created a redirect for Shah function to here.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
4224:
3828:
2361:{\displaystyle \int _{f_{0}}^{f_{0}+1/T}e^{i2\pi mfT}e^{-i2\pi nfT}df={\frac {1}{T}}\delta _{mn}}
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966:). your "shah" function is just three letters: "I" and what we should really get is the
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3551:. Also, the (unitary) Fourier transform of the comb was incorrect. The period is
2492:{\displaystyle c_{n}={\frac {1}{T}}\int _{f_{0}}^{f_{0}+1/T}x(t)e^{-i2\pi nfT}df}
3256:{\displaystyle 1\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ \delta (f).}
1242:
1069:
1018:
The only way I've found to make the symbol is to use three upright 'I' letters:
207:
102:
406:
article which have the same symbol with subscript as the former symbol for the
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157:
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3494:
1785:{\displaystyle x(t)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }\delta (t-kT)\ }
423:
407:
184:
79:
3751:{\displaystyle \delta (\sin x)=\sum _{n=-\infty }^{\infty }\delta (t-n\pi ),}
1608:{\displaystyle c_{n}=f_{0}\int _{t_{0}}^{t_{0}+T}x(t)e^{-i2\pi nf_{0}t}dt\ }
422:
oh, and i forgot to mention that i looked up all of the articles linked to
2084:{\displaystyle x(f)=\sum _{k=-\infty }^{\infty }\delta (f-{\frac {k}{T}})}
3918:
3917:
I restored the scaling property, expressed properly, with justification.
3869:
3808:
1396:{\displaystyle x(t)=\sum _{n=-\infty }^{\infty }c_{n}e^{i2\pi nf_{0}t}\ }
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Now this needs source or proof. Is it really a
Schwartz distribution?
988:) and add Oppenheim and Schafer to the reference list? Where does the
4233:
4139:
level of assumed knowledge is inescapable (we're not going to define
2096:
1909:{\displaystyle x(t)=\sum _{n=-\infty }^{\infty }e^{i2\pi nf_{0}t}\ }
2185:{\displaystyle x(f)=\sum _{n=-\infty }^{\infty }c_{n}e^{i2\pi nfT}}
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that is in the range we are integrating (-1/2T,1/2T) over is with
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Dirac_delta_function#Delta_function_of_more_complicated_arguments
4234:
Shouldn't we inform readers that the imaginary terms cancel out?
3477:{\displaystyle F(\nu )=\delta (\nu ).\quad \quad {\mbox{QED}}\,}
410:. two qualitatively different functions, same symbol --- bad.
402:
to differentiate it from the "nascent" impulse functions in the
1068:
However, the hinting causes these 'I's to have odd spacing. --
15:
3265:
Then just write out the Poisson summation formula (row 602,
3228:
873:
The article refers to Bracewell and he uses the shah-symbol
3267:
Relationship between functions and their Fourier Transforms
845:
knock yourself out. there is a mention of this scaling at
2726:
is periodic over 1/T, we can choose a simple range with
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206:, a collaborative effort to improve the coverage of
101:, a collaborative effort to improve the coverage of
426:and fixed any reference there so we're consistent.
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750:can be obtained by an appropriate scaling of the
1675:{\displaystyle -\infty <t_{0}<+\infty \ }
8:
3864:in linear statistics is usually taken to be
1458:{\displaystyle f_{0}\equiv {\frac {1}{T}}\ }
3978:The very first equation already introduces
19:
3938:
3203:I assume you're OK with the transform pair
352:i changed the symbol for the Dirac comb...
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47:
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2015:
1959:
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993:
933:
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878:
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792:
786:
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759:
734:
728:
723:which means that any Dirac comb function
699:
687:
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661:
641:
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612:
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586:
563:
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494:
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288:
4118:, by implication is time of course, and
3472:
154:
49:
4219:here are not in that category, IMO. --
3890:I have to fully agree with the IP who
339:Thanks, that's what I meant to do. --
4240:2601:200:C000:1A0:60BA:5A24:E2E8:77BF
3886:Deletion of Scaling property section.
1057:{\displaystyle \mathrm {III} _{T}(t)}
7:
3892:deleted the Scaling property section
3761:in case if someone is interested...
2860:{\displaystyle c_{n}={\frac {1}{T}}}
808:from which the Fourier transform of
200:This article is within the scope of
95:This article is within the scope of
4277:Low-importance Engineering articles
3868:=exp(θ), which has a period of 2π.
1975:{\displaystyle e^{i2\pi nf_{0}t}\ }
444:scaling property is the following:
38:It is of interest to the following
4179:in this context for example). But
4146:
3719:
3714:
3360:
3355:
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2137:
2051:
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1867:
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1344:
1339:
1198:
1193:
1140:
1135:
1035:
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1029:
995:
935:
915:
816:
789:
762:
731:
684:
627:
622:
543:
538:
495:
490:
455:
372:
317:{\displaystyle \sum \delta (t-nT)}
267:This is the same as the so-called
14:
4267:Low-priority mathematics articles
3610:{\displaystyle {\sqrt {2\pi }}/T}
220:Knowledge:WikiProject Engineering
115:Knowledge:WikiProject Mathematics
4282:WikiProject Engineering articles
4272:Start-Class Engineering articles
4262:Start-Class mathematics articles
3959:Added "citation needed" tag. --
1927:Then ask yourself: What is the
395:{\displaystyle \Delta _{T}(t)\ }
223:Template:WikiProject Engineering
187:
177:
156:
118:Template:WikiProject Mathematics
82:
72:
51:
20:
240:This article has been rated as
135:This article has been rated as
4229:04:11, 20 September 2016 (UTC)
3742:
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3503:17:19, 16 September 2008 (UTC)
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3449:
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3015:Since you have no problem with
3007:07:44, 16 September 2008 (UTC)
2922:
2903:
2719:{\displaystyle e^{-i2\pi nfT}}
2625:
2606:
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2117:
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2020:
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280:I think so - I don't know if
1:
3968:13:07, 26 December 2014 (UTC)
3953:12:17, 26 December 2014 (UTC)
3927:04:33, 27 December 2011 (UTC)
3912:03:59, 27 December 2011 (UTC)
3878:22:14, 25 November 2010 (UTC)
3833:15:12, 25 November 2010 (UTC)
3817:09:48, 18 November 2010 (UTC)
3793:17:06, 21 November 2009 (UTC)
3463:
3461:
2869:Therefore we have shown that
1174:
1109:
1093:00:31, 26 February 2020 (UTC)
214:and see a list of open tasks.
109:and see a list of open tasks.
3772:23:42, 28 October 2009 (UTC)
3638:19:37, 23 October 2013 (UTC)
2008:First look at the function:
4098:should be defined as well.
2766:{\displaystyle f_{0}=-1/2T}
828:{\displaystyle \Delta _{T}}
801:{\displaystyle \Delta _{1}}
774:{\displaystyle \Delta _{1}}
743:{\displaystyle \Delta _{T}}
344:15:41, 27 August 2005 (UTC)
332:21:27, 26 August 2005 (UTC)
276:20:23, 26 August 2005 (UTC)
4298:
2997:which is the desired eqn.
1013:22:36, 2 August 2006 (UTC)
975:21:05, 2 August 2006 (UTC)
953:21:00, 2 August 2006 (UTC)
864:21:02, 2 August 2006 (UTC)
854:20:59, 2 August 2006 (UTC)
840:21:02, 2 August 2006 (UTC)
835:easily can be derived. --
246:project's importance scale
3488:Poisson summation formula
3424:explicitly for the case
1995:20:05, 12 June 2007 (UTC)
1246:18:52, 12 June 2007 (UTC)
1078:01:36, 10 June 2013 (UTC)
239:
172:
134:
67:
46:
4248:03:33, 7 July 2022 (UTC)
3660:14:41, 4 July 2009 (UTC)
1809:. You'll find that all
431:05:45, 8 June 2006 (UTC)
415:05:44, 8 June 2006 (UTC)
141:project's priority scale
4192:{\displaystyle \delta }
4152:{\displaystyle \Sigma }
4131:{\displaystyle \delta }
4071:{\displaystyle \delta }
3991:{\displaystyle \delta }
3575:{\displaystyle 2\pi /T}
3544:{\displaystyle \omega }
2773:, so the only value of
2196:where the coefficients
1261:) be periodic, so that
1001:{\displaystyle \Delta }
941:{\displaystyle \Delta }
921:{\displaystyle \Delta }
203:WikiProject Engineering
98:WikiProject Mathematics
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3933:Schwartz distribution?
3799:Directional statistics
3752:
3723:
3669:It can be shown that
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1144:
1058:
1008:symbol comes from? --
1002:
942:
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902:
901:{\displaystyle III(t)}
829:
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318:
28:This article is rated
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2216:{\displaystyle c_{n}}
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3896:contributed long ago
3853:wrapped distribution
3805:wrapped distribution
3779:Dirac delta function
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754:Dirac comb function
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404:Dirac delta function
368:
287:
226:Engineering articles
121:mathematics articles
3974:Terms need defining
3140:
3073:
2820:{\displaystyle k=0}
2794:{\displaystyle k/T}
2573:
2442:
2272:
1549:
4209:
4189:
4169:
4149:
4128:
4108:
4088:
4068:
4048:
4028:
4008:
3988:
3748:
3607:
3572:
3541:
3521:
3490:is worth knowing.
3474:
3473:
3470:
3464:
3462:
3410:
3253:
3190:
3123:
3034:
2987:
2857:
2817:
2791:
2763:
2716:
2667:
2531:
2489:
2400:
2358:
2230:
2213:
2182:
2081:
1972:
1906:
1782:
1672:
1605:
1515:
1455:
1393:
1231:
1175:
1110:
1054:
998:
938:
918:
898:
825:
798:
771:
740:
710:
392:
314:
195:Engineering portal
90:Mathematics portal
34:content assessment
4212:{\displaystyle t}
4172:{\displaystyle k}
4111:{\displaystyle t}
4091:{\displaystyle t}
4051:{\displaystyle T}
4031:{\displaystyle T}
4011:{\displaystyle t}
3966:
3955:
3943:comment added by
3641:
3624:comment added by
3597:
3524:{\displaystyle f}
3486:The proof of the
3469:
3381:
3328:
3237:
3233:
3217:
3071:
3055:
2936:
2920:
2855:
2681:Remembering that
2623:
2529:
2398:
2343:
2076:
1971:
1929:Fourier Transform
1905:
1781:
1733:
1671:
1604:
1454:
1451:
1392:
1167:
1119:
1098:Fourier transform
681:
606:
391:
260:
259:
256:
255:
252:
251:
151:
150:
147:
146:
4289:
4218:
4216:
4215:
4210:
4198:
4196:
4195:
4190:
4178:
4176:
4175:
4170:
4158:
4156:
4155:
4150:
4137:
4135:
4134:
4129:
4117:
4115:
4114:
4109:
4097:
4095:
4094:
4089:
4077:
4075:
4074:
4069:
4057:
4055:
4054:
4049:
4037:
4035:
4034:
4029:
4017:
4015:
4014:
4009:
3997:
3995:
3994:
3989:
3963:
3785:
3766:
3757:
3755:
3754:
3749:
3722:
3717:
3640:
3618:
3616:
3614:
3613:
3608:
3603:
3598:
3590:
3581:
3579:
3578:
3573:
3568:
3550:
3548:
3547:
3542:
3530:
3528:
3527:
3522:
3483:
3481:
3480:
3475:
3471:
3467:
3419:
3417:
3416:
3411:
3409:
3408:
3380:
3363:
3358:
3334:
3330:
3329:
3321:
3304:
3299:
3262:
3260:
3259:
3254:
3236:
3235:
3234:
3232:
3231:
3225:
3220:
3216:
3199:
3197:
3196:
3191:
3180:
3179:
3139:
3134:
3113:
3112:
3072:
3070:
3059:
3057:
3056:
3054:
3043:
2996:
2994:
2993:
2988:
2986:
2985:
2960:
2955:
2937:
2929:
2921:
2913:
2898:
2893:
2866:
2864:
2863:
2858:
2856:
2848:
2843:
2842:
2826:
2824:
2823:
2818:
2800:
2798:
2797:
2792:
2787:
2772:
2770:
2769:
2764:
2756:
2742:
2741:
2725:
2723:
2722:
2717:
2715:
2714:
2676:
2674:
2673:
2668:
2660:
2659:
2632:
2628:
2624:
2616:
2601:
2596:
2572:
2568:
2557:
2556:
2546:
2545:
2544:
2530:
2522:
2517:
2516:
2498:
2496:
2495:
2490:
2482:
2481:
2441:
2437:
2426:
2425:
2415:
2414:
2413:
2399:
2391:
2386:
2385:
2367:
2365:
2364:
2359:
2357:
2356:
2344:
2336:
2325:
2324:
2297:
2296:
2271:
2267:
2256:
2255:
2245:
2244:
2243:
2222:
2220:
2219:
2214:
2212:
2211:
2191:
2189:
2188:
2183:
2181:
2180:
2156:
2155:
2145:
2140:
2090:
2088:
2087:
2082:
2077:
2069:
2054:
2049:
1981:
1979:
1978:
1973:
1970:
1969:
1968:
1964:
1963:
1915:
1913:
1912:
1907:
1904:
1903:
1902:
1898:
1897:
1870:
1865:
1791:
1789:
1788:
1783:
1780:
1757:
1752:
1734:
1726:
1682:can be anything.
1681:
1679:
1678:
1673:
1670:
1660:
1659:
1614:
1612:
1611:
1606:
1603:
1596:
1595:
1591:
1590:
1548:
1541:
1540:
1530:
1529:
1528:
1514:
1513:
1501:
1500:
1464:
1462:
1461:
1456:
1453:
1452:
1444:
1439:
1438:
1402:
1400:
1399:
1394:
1391:
1390:
1389:
1385:
1384:
1358:
1357:
1347:
1342:
1240:
1238:
1237:
1232:
1230:
1229:
1201:
1196:
1173:
1169:
1168:
1160:
1143:
1138:
1120:
1112:
1063:
1061:
1060:
1055:
1044:
1043:
1038:
1007:
1005:
1004:
999:
947:
945:
944:
939:
927:
925:
924:
919:
907:
905:
904:
899:
834:
832:
831:
826:
824:
823:
807:
805:
804:
799:
797:
796:
780:
778:
777:
772:
770:
769:
749:
747:
746:
741:
739:
738:
719:
717:
716:
711:
703:
692:
691:
682:
680:
679:
671:
662:
645:
630:
625:
607:
605:
604:
596:
587:
567:
546:
541:
498:
493:
463:
462:
439:Scaling property
401:
399:
398:
393:
390:
380:
379:
323:
321:
320:
315:
228:
227:
224:
221:
218:
197:
192:
191:
181:
174:
173:
168:
160:
153:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
4297:
4296:
4292:
4291:
4290:
4288:
4287:
4286:
4252:
4251:
4236:
4201:
4200:
4181:
4180:
4161:
4160:
4141:
4140:
4120:
4119:
4100:
4099:
4080:
4079:
4060:
4059:
4040:
4039:
4020:
4019:
4000:
3999:
3980:
3979:
3976:
3935:
3888:
3801:
3783:
3764:
3674:
3673:
3667:
3647:
3619:
3584:
3583:
3553:
3552:
3533:
3532:
3513:
3512:
3426:
3425:
3382:
3313:
3309:
3277:
3276:
3208:
3207:
3153:
3086:
3063:
3047:
3023:
3022:
2962:
2871:
2870:
2834:
2829:
2828:
2803:
2802:
2775:
2774:
2733:
2728:
2727:
2688:
2683:
2682:
2633:
2578:
2574:
2548:
2536:
2508:
2503:
2502:
2455:
2417:
2405:
2377:
2372:
2371:
2345:
2298:
2273:
2247:
2235:
2225:
2224:
2203:
2198:
2197:
2157:
2147:
2103:
2102:
2012:
2011:
1992:207.190.198.135
1955:
1938:
1933:
1932:
1889:
1872:
1828:
1827:
1814:
1807:
1705:
1704:
1651:
1637:
1636:
1582:
1562:
1532:
1520:
1505:
1492:
1487:
1486:
1430:
1425:
1424:
1376:
1359:
1349:
1305:
1304:
1241:? Thank you. --
1203:
1152:
1148:
1104:
1103:
1100:
1027:
1022:
1021:
990:
989:
930:
929:
910:
909:
875:
874:
871:
869:The shah symbol
815:
810:
809:
788:
783:
782:
761:
756:
755:
730:
725:
724:
683:
666:
591:
454:
449:
448:
441:
371:
366:
365:
359:
354:
285:
284:
265:
225:
222:
219:
216:
215:
193:
186:
166:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
4295:
4293:
4285:
4284:
4279:
4274:
4269:
4264:
4254:
4253:
4235:
4232:
4208:
4188:
4168:
4148:
4127:
4107:
4087:
4067:
4047:
4038:defining only
4027:
4007:
3987:
3975:
3972:
3971:
3970:
3934:
3931:
3930:
3929:
3904:71.169.185.162
3887:
3884:
3883:
3882:
3881:
3880:
3845:
3844:
3843:
3842:
3836:
3835:
3800:
3797:
3796:
3795:
3784:Sławomir Biały
3765:Zitterbewegung
3759:
3758:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3721:
3716:
3713:
3710:
3707:
3703:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3666:
3663:
3652:Sławomir Biały
3646:
3643:
3606:
3602:
3596:
3593:
3571:
3567:
3563:
3560:
3540:
3520:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3423:
3421:
3420:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3385:
3379:
3376:
3373:
3370:
3367:
3362:
3357:
3354:
3351:
3348:
3344:
3340:
3337:
3333:
3327:
3324:
3319:
3316:
3312:
3308:
3303:
3298:
3295:
3292:
3289:
3285:
3252:
3249:
3246:
3243:
3240:
3230:
3224:
3215:
3201:
3200:
3189:
3186:
3183:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3156:
3152:
3149:
3146:
3143:
3138:
3133:
3130:
3126:
3122:
3119:
3116:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3089:
3085:
3082:
3079:
3076:
3069:
3066:
3062:
3053:
3050:
3046:
3041:
3037:
3033:
3030:
3014:
3012:
3011:
3010:
3009:
2984:
2981:
2978:
2975:
2972:
2969:
2965:
2959:
2954:
2951:
2948:
2945:
2941:
2935:
2932:
2927:
2924:
2919:
2916:
2911:
2908:
2905:
2902:
2897:
2892:
2889:
2886:
2883:
2879:
2867:
2854:
2851:
2846:
2841:
2837:
2816:
2813:
2810:
2790:
2786:
2782:
2762:
2759:
2755:
2751:
2748:
2745:
2740:
2736:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2691:
2679:
2678:
2677:
2666:
2663:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2636:
2631:
2627:
2622:
2619:
2614:
2611:
2608:
2605:
2600:
2595:
2592:
2589:
2586:
2582:
2577:
2571:
2567:
2563:
2560:
2555:
2551:
2543:
2539:
2534:
2528:
2525:
2520:
2515:
2511:
2500:
2488:
2485:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2458:
2454:
2451:
2448:
2445:
2440:
2436:
2432:
2429:
2424:
2420:
2412:
2408:
2403:
2397:
2394:
2389:
2384:
2380:
2355:
2352:
2348:
2342:
2339:
2334:
2331:
2328:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2301:
2295:
2292:
2289:
2286:
2283:
2280:
2276:
2270:
2266:
2262:
2259:
2254:
2250:
2242:
2238:
2233:
2210:
2206:
2194:
2193:
2192:
2179:
2176:
2173:
2170:
2167:
2164:
2160:
2154:
2150:
2144:
2139:
2136:
2133:
2130:
2126:
2122:
2119:
2116:
2113:
2110:
2093:
2092:
2091:
2080:
2075:
2072:
2067:
2064:
2061:
2058:
2053:
2048:
2045:
2042:
2039:
2035:
2031:
2028:
2025:
2022:
2019:
2006:
2000:
1999:
1998:
1997:
1986:
1985:
1984:
1983:
1967:
1962:
1958:
1954:
1951:
1948:
1945:
1941:
1922:
1921:
1920:
1919:
1918:
1917:
1901:
1896:
1892:
1888:
1885:
1882:
1879:
1875:
1869:
1864:
1861:
1858:
1855:
1851:
1847:
1844:
1841:
1838:
1835:
1820:
1819:
1818:
1817:
1812:
1805:
1802:and solve for
1797:
1796:
1795:
1794:
1793:
1792:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1756:
1751:
1748:
1745:
1742:
1738:
1732:
1729:
1724:
1721:
1718:
1715:
1712:
1697:
1696:
1695:
1694:
1688:
1687:
1686:
1685:
1684:
1683:
1669:
1666:
1663:
1658:
1654:
1650:
1647:
1644:
1629:
1628:
1627:
1626:
1620:
1619:
1618:
1617:
1616:
1615:
1602:
1599:
1594:
1589:
1585:
1581:
1578:
1575:
1572:
1569:
1565:
1561:
1558:
1555:
1552:
1547:
1544:
1539:
1535:
1527:
1523:
1518:
1512:
1508:
1504:
1499:
1495:
1479:
1478:
1477:
1476:
1470:
1469:
1468:
1467:
1466:
1465:
1450:
1447:
1442:
1437:
1433:
1417:
1416:
1415:
1414:
1408:
1407:
1406:
1405:
1404:
1403:
1388:
1383:
1379:
1375:
1372:
1369:
1366:
1362:
1356:
1352:
1346:
1341:
1338:
1335:
1332:
1328:
1324:
1321:
1318:
1315:
1312:
1297:
1296:
1295:
1294:
1283:Fourier series
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1206:
1200:
1195:
1192:
1189:
1186:
1182:
1178:
1172:
1166:
1163:
1158:
1155:
1151:
1147:
1142:
1137:
1134:
1131:
1128:
1124:
1118:
1115:
1099:
1096:
1081:
1080:
1066:
1065:
1064:
1053:
1050:
1047:
1042:
1037:
1034:
1031:
997:
978:
977:
937:
917:
897:
894:
891:
888:
885:
882:
870:
867:
857:
856:
822:
818:
795:
791:
768:
764:
737:
733:
721:
720:
709:
706:
702:
698:
695:
690:
686:
678:
674:
670:
665:
660:
657:
654:
651:
648:
644:
640:
637:
634:
629:
624:
621:
618:
615:
611:
603:
599:
595:
590:
585:
582:
579:
576:
573:
570:
566:
562:
559:
556:
553:
550:
545:
540:
537:
534:
531:
527:
523:
520:
517:
514:
511:
508:
505:
502:
497:
492:
489:
486:
483:
479:
475:
472:
469:
466:
461:
457:
440:
437:
436:
435:
434:
433:
389:
386:
383:
378:
374:
357:
353:
350:
349:
348:
347:
346:
325:
324:
313:
310:
307:
304:
301:
298:
295:
292:
264:
261:
258:
257:
254:
253:
250:
249:
242:Low-importance
238:
232:
231:
229:
212:the discussion
199:
198:
182:
170:
169:
167:Low‑importance
161:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
4294:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4259:
4257:
4250:
4249:
4245:
4241:
4231:
4230:
4226:
4222:
4206:
4186:
4166:
4125:
4105:
4085:
4065:
4045:
4025:
4005:
3985:
3973:
3969:
3965:
3962:
3958:
3957:
3956:
3954:
3950:
3946:
3942:
3932:
3928:
3924:
3920:
3916:
3915:
3914:
3913:
3909:
3905:
3901:
3897:
3893:
3885:
3879:
3875:
3871:
3867:
3863:
3859:
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3653:
3645:Bed of nails?
3644:
3642:
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3631:
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3604:
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3594:
3591:
3569:
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3458:
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3431:
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3402:
3399:
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3349:
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3338:
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3325:
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3314:
3310:
3306:
3293:
3290:
3287:
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3275:
3274:
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3263:
3250:
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3238:
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3141:
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3106:
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3100:
3097:
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3091:
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3064:
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3039:
3035:
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3028:
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3008:
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3000:
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2231:
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2204:
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2171:
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2165:
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2152:
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2124:
2120:
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2108:
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2100:
2099:of the form:
2098:
2094:
2073:
2070:
2065:
2062:
2056:
2043:
2040:
2037:
2033:
2029:
2023:
2017:
2010:
2009:
2007:
2004:
2003:
2002:
2001:
1996:
1993:
1990:
1989:
1988:
1987:
1965:
1960:
1956:
1952:
1949:
1946:
1943:
1939:
1930:
1926:
1925:
1924:
1923:
1899:
1894:
1890:
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1859:
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1824:
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1800:
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1423:
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1419:
1418:
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1367:
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1354:
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1336:
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1326:
1322:
1316:
1310:
1303:
1302:
1301:
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1299:
1298:
1292:
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1226:
1223:
1220:
1217:
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1129:
1126:
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1095:
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1075:
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1020:
1019:
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969:
965:
961:
957:
956:
955:
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883:
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616:
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506:
500:
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481:
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467:
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446:
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420:
419:
418:
417:
416:
413:
409:
405:
384:
376:
363:
351:
345:
342:
338:
337:
336:
335:
334:
333:
330:
308:
305:
302:
299:
293:
290:
283:
282:
281:
278:
277:
274:
270:
269:Shah function
263:Shah function
262:
247:
243:
237:
234:
233:
230:
213:
209:
205:
204:
196:
190:
185:
183:
180:
176:
175:
171:
165:
162:
159:
155:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
4237:
3977:
3945:92.52.10.154
3939:— Preceding
3936:
3899:
3889:
3865:
3861:
3857:
3802:
3762:
3760:
3668:
3648:
3620:— Preceding
3509:User:rdsomma
3506:
3492:
3485:
3422:
3270:
3264:
3204:
3202:
3016:
3013:
2827:, giving us
1810:
1803:
1290:
1286:
1281:. Then the
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1101:
1082:
985:
981:
979:
963:
959:
872:
858:
751:
722:
442:
361:
355:
326:
279:
271:, right? --
268:
266:
241:
201:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
2999:24.58.159.5
1816:= 1 so that
217:Engineering
208:engineering
164:Engineering
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
4256:Categories
3894:. It was
1277:) for all
752:normalized
424:Dirac comb
408:Dirac comb
2368:) to get
1625:and where
4221:Bwechner
3941:unsigned
3825:Melcombe
3634:contribs
3622:unsigned
1085:cherkash
970:symbol.
968:cyrillic
356:... to Δ
341:Pgabolde
273:Pgabolde
3626:Rdsomma
3582:, not
1693:Now set
244:on the
139:on the
3665:A fact
2097:ansatz
1243:Abdull
1070:Tweenk
36:scale.
3961:intgr
3900:could
3495:Bob K
1413:where
972:r b-j
851:r b-j
428:r b-j
412:r b-j
364:) or
4244:talk
4225:talk
4199:and
4078:and
4018:and
3949:talk
3923:talk
3908:talk
3874:talk
3829:talk
3813:talk
3789:talk
3656:talk
3630:talk
3499:talk
3003:talk
1662:<
1649:<
1293:) is
1285:for
1269:) =
1253:Let
1102:Why
1089:talk
1074:talk
4159:or
3919:PAR
3870:PAR
3809:PAR
3686:sin
3617:.
3531:or
3468:QED
1931:of
1475:and
1275:t-T
1010:KYN
950:KYN
861:KYN
837:KYN
329:PAR
236:Low
131:Low
4258::
4246:)
4227:)
4187:δ
4147:Σ
4126:δ
4066:δ
4058:.
3998:,
3986:δ
3951:)
3925:)
3910:)
3876:)
3831:)
3815:)
3791:)
3740:π
3734:−
3725:δ
3720:∞
3715:∞
3712:−
3702:∑
3689:
3680:δ
3658:)
3636:)
3632:•
3595:π
3562:π
3539:ω
3507:--
3501:)
3493:--
3453:ν
3447:δ
3438:ν
3406:ν
3397:π
3388:−
3361:∞
3356:∞
3353:−
3343:∑
3315:ν
3302:∞
3297:∞
3294:−
3284:∑
3239:δ
3223:⟺
3168:π
3159:−
3142:δ
3137:∞
3132:∞
3129:−
3125:∫
3101:π
3092:−
3075:δ
3040:−
3036:∫
3005:)
2974:π
2958:∞
2953:∞
2950:−
2940:∑
2910:−
2901:δ
2896:∞
2891:∞
2888:−
2878:∑
2747:−
2703:π
2694:−
2648:π
2639:−
2613:−
2604:δ
2599:∞
2594:∞
2591:−
2581:∑
2533:∫
2499:or
2470:π
2461:−
2402:∫
2347:δ
2313:π
2304:−
2285:π
2232:∫
2169:π
2143:∞
2138:∞
2135:−
2125:∑
2066:−
2057:δ
2052:∞
2047:∞
2044:−
2034:∑
1950:π
1884:π
1868:∞
1863:∞
1860:−
1850:∑
1769:−
1760:δ
1755:∞
1750:∞
1747:−
1737:∑
1668:∞
1646:∞
1643:−
1577:π
1568:−
1517:∫
1441:≡
1371:π
1345:∞
1340:∞
1337:−
1327:∑
1218:π
1209:−
1199:∞
1194:∞
1191:−
1181:∑
1157:−
1146:δ
1141:∞
1136:∞
1133:−
1123:∑
1091:)
1076:)
996:Δ
936:Δ
916:Δ
849:.
817:Δ
790:Δ
763:Δ
732:Δ
685:Δ
650:−
633:δ
628:∞
623:∞
620:−
610:∑
572:−
549:δ
544:∞
539:∞
536:−
526:∑
510:−
501:δ
496:∞
491:∞
488:−
478:∑
456:Δ
373:Δ
303:−
294:δ
291:∑
4242:(
4223:(
4207:t
4167:k
4106:t
4086:t
4046:T
4026:T
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3947:(
3921:(
3906:(
3872:(
3866:z
3862:x
3858:x
3827:(
3811:(
3787:(
3746:,
3743:)
3737:n
3731:t
3728:(
3709:=
3706:n
3698:=
3695:)
3692:x
3683:(
3654:(
3628:(
3605:T
3601:/
3592:2
3570:T
3566:/
3559:2
3519:f
3497:(
3459:.
3456:)
3450:(
3444:=
3441:)
3435:(
3432:F
3403:T
3400:n
3394:2
3391:i
3384:e
3378:)
3375:T
3372:n
3369:(
3366:f
3350:=
3347:n
3339:T
3336:=
3332:)
3326:T
3323:k
3318:+
3311:(
3307:F
3291:=
3288:k
3271::
3269:)
3251:.
3248:)
3245:f
3242:(
3229:F
3214:1
3205::
3188:,
3185:f
3182:d
3177:T
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3165:2
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3155:e
3151:)
3148:f
3145:(
3121:=
3118:f
3115:d
3110:T
3107:f
3104:n
3098:2
3095:i
3088:e
3084:)
3081:f
3078:(
3068:T
3065:2
3061:1
3052:T
3049:2
3045:1
3032:=
3029:1
3017::
3001:(
2983:T
2980:f
2977:n
2971:2
2968:i
2964:e
2947:=
2944:n
2934:T
2931:1
2926:=
2923:)
2918:T
2915:k
2907:f
2904:(
2885:=
2882:k
2853:T
2850:1
2845:=
2840:n
2836:c
2815:0
2812:=
2809:k
2789:T
2785:/
2781:k
2761:T
2758:2
2754:/
2750:1
2744:=
2739:0
2735:f
2712:T
2709:f
2706:n
2700:2
2697:i
2690:e
2665:f
2662:d
2657:T
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2651:n
2645:2
2642:i
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2630:)
2626:)
2621:T
2618:k
2610:f
2607:(
2588:=
2585:k
2576:(
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2566:/
2562:1
2559:+
2554:0
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2524:1
2519:=
2514:n
2510:c
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2484:d
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2453:)
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2435:/
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2388:=
2383:n
2379:c
2354:n
2351:m
2341:T
2338:1
2333:=
2330:f
2327:d
2322:T
2319:f
2316:n
2310:2
2307:i
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2294:T
2291:f
2288:m
2282:2
2279:i
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2269:T
2265:/
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2209:n
2205:c
2178:T
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2172:n
2166:2
2163:i
2159:e
2153:n
2149:c
2132:=
2129:n
2121:=
2118:)
2115:f
2112:(
2109:x
2079:)
2074:T
2071:k
2063:f
2060:(
2041:=
2038:k
2030:=
2027:)
2024:f
2021:(
2018:x
1982:?
1966:t
1961:0
1957:f
1953:n
1947:2
1944:i
1940:e
1916:.
1900:t
1895:0
1891:f
1887:n
1881:2
1878:i
1874:e
1857:=
1854:n
1846:=
1843:)
1840:t
1837:(
1834:x
1813:n
1811:c
1806:n
1804:c
1778:)
1775:T
1772:k
1766:t
1763:(
1744:=
1741:k
1731:T
1728:1
1723:=
1720:)
1717:t
1714:(
1711:x
1665:+
1657:0
1653:t
1601:t
1598:d
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1554:(
1551:x
1546:T
1543:+
1538:0
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1526:0
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1503:=
1498:n
1494:c
1449:T
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1432:f
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1382:0
1378:f
1374:n
1368:2
1365:i
1361:e
1355:n
1351:c
1334:=
1331:n
1323:=
1320:)
1317:t
1314:(
1311:x
1291:t
1289:(
1287:x
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1273:(
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1265:(
1263:x
1259:t
1257:(
1255:x
1227:T
1224:n
1221:f
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1205:e
1188:=
1185:n
1177:=
1171:)
1165:T
1162:k
1154:f
1150:(
1130:=
1127:k
1117:T
1114:1
1087:(
1072:(
1052:)
1049:t
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986:t
984:(
982:s
964:t
962:(
960:s
896:)
893:t
890:(
887:I
884:I
881:I
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794:1
767:1
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708:)
705:T
701:/
697:t
694:(
689:1
677:|
673:T
669:|
664:1
659:=
656:)
653:k
647:T
643:/
639:t
636:(
617:=
614:k
602:|
598:T
594:|
589:1
584:=
581:)
578:]
575:k
569:T
565:/
561:t
558:[
555:T
552:(
533:=
530:k
522:=
519:)
516:T
513:k
507:t
504:(
485:=
482:k
474:=
471:)
468:t
465:(
460:T
388:)
385:t
382:(
377:T
362:t
360:(
358:T
312:)
309:T
306:n
300:t
297:(
248:.
143:.
42::
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