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According to functions.wolfram.com, the zeros of the Airy function $\operatorname{Ai}(z)$ occur at $z_k=f\left(\tfrac{3\pi}{8}(4k-1)\right)$ for $k\in \mathbb{N}$ where

$f(d)=-d^{2/3} \left(1 + \frac{5}{48 d^2} - \frac{5}{36 d^4} + \frac{77125}{82944 d^6} - \frac{108056875}{6967296 d^8} + \frac{162375596875}{334430208 d^{10}} - \frac{1622671914671875}{66217181184 d^{12}} + \frac{150126478779573265625}{82639042117632 d^{14}} - \frac{644932726927939889453125}{3470839768940544 d^{16}} + \frac{13042116997445589075044921875}{520200964553048064 d^{18}} - \frac{569789860268573944980176052734375}{132083753999696658432 d^{20}}\right)$

At first I thought this was amazing yet plausible, as I had recently learned about how the integer solutions to the Airy zeta function take on nice polynomial forms -- see for example this mathematics SE or this mathworld.wolfram.com entry. However, when I actually tried to evaluate $f(d)$ in Mathematica 12.3, I got a different answer.

f[d_] := -d^(2/3)*(1 + 5/(48*d^2) - 5/(36*d^4) + 77125/(82944*d^6) - 
     108056875/(6967296*d^8) + 162375596875/(334430208*d^10) - 
     1622671914671875/(66217181184*d^12) + 
     150126478779573265625/(82639042117632*d^14) - 
     644932726927939889453125/(3470839768940544*d^16) + 
     13042116997445589075044921875/(520200964553048064*d^18) - 
     569789860268573944980176052734375/(132083753999696658432*d^20));

Table[N[f[3 \[Pi] (4 k - 1)/8] - AiryAiZero[k], 300], {k, 1, 10}]

Table[N[AiryAi[f[3 \[Pi] (4 k - 1)/8]], 300], {k, 1, 10}]

Result:

{101.024729729643517052210195897494048933937448846959317979056170305713556033155709344313873339785588316724623893628444173954660751781777382909884227119991063653922489122521028035507326422246820617581835086039448182279716735931878191831471705993891347564944216289890092732684563643492816815687014128001,
6.07199647755491716780820663162350058052129033624599025089153818243790305756246446381129835095260678865742048597175215792151178403339439669120154396823806647834859731660635713774747809505012472067936314895844551557352019151100045768677814067792800572579483822873474708680322176881858662601218743043280*10^-6,
7.10693118053166874253092860683832714062926451017745243810780402099092972962760290471065683614368438966158861098673688402608772707984582970755578456058116507577755571593281846774255904063966424239121507655265657009205320931193150786681946510836085339892733855591162255870686794266507295425938644582085*10^-10,
1.28489459569314906942663751018874961963048899050599323001698957185114246278003914920064047534980202400138241663422793129859141031067482070291698627873816526587887879111930526211284432872189498293584447033237738920880381607802936681010978845674080167330406956463956297523687507306691832671238923137211*10^-12,
9.85230521408823532334323017144899699093858694829466757917449600281060176469994829311163861296342767315301178511076803198740252811023393771944999784070991886897637574706385889683700098710168257514900435009534840690895047886758424119559569106358885733288039855378350050463050710635946912715323606519969*10^-15,
1.86059836921457379993283531720867666774198098186579688206016954203264076839572030475548228986146982409440096743573237423546792068413701518038094506231348473805824916533584194933354979953918048600376071479494602417967698605441485223869125110703202928930266114385103470438607932317199426139389840002945*10^-16,
6.52206523462147938566440091394091215532843766432777114805082045041388329884095395841174130265618498546297259277621066039953029393992128000280031999001088571561366344135198642142689147191178752878633072548229637652565578219008625484797231698915856536749228530541818874862161243350886223355472220037127*10^-18,
3.59112912990885327022402699248768594493008198186326985854905140103891567761151859282551424533891171904338489175108513993380501901013115089835887712412188553140985516880029957983269953644579627972623793138255386239719613290960898482530313947660261355369419475157148326900695137602838908469622343840893*10^-19,
2.79036445820173811525072394862106114138144456589315694914070080050894389307895249135042487608925823118479871253803460358490242094448702997004059757765155116619137135327926032813404265920795750478596593989040413404451262883200435972541883559460479259083162259885315690364742616882249503091450423360600*10^-20,
2.84433039717560102652822420096934352340238918157480259506581097615889232292099637932479751597061473971694682966465883110136644734963760199559005658526992189043579481244083889458147877969249225350250298508072702695199535036569392354131250510224221516997206728659621034255684269858727948199678819835416*10^-21}

{1.28022662320311490464244954298064452767794276939650182506247332190299068692033876546194413030648678485416280393825673826416210147906981938129034010839350867015605213197214594084365865408745823401542424992401529676186815380466907099143978523621115786847790693704717923827700925592231232219378135284984*10^-285,
-4.87648940750614231893035097087567012614252919960419557989830809999591199286314354989632208751741079918422835822898078088552266004163560087594847325635005204961558256602017881812604036362250037931943200185223357818586986419884600771831041739798812982796350933012648114353288383504305428150412493689887*10^-6,
6.14894546914867767058728407137545539599857058592749046260957873915316075113149465251290700790198016004712457998001935270555346125777943275122526690175356719878285950371094032485136154416230018174673723490746828333500060731036067049023778707139427660638121850997721719469094448262430971902630470250812*10^-10,
-1.17034718951815494444408687099641554112757498631095818326833069475269475868196228420775869478055386957375835377896446643200798223415511325074474962042363171185316397399557001180703198077679498189814625784796662072354924019598629813888925177575830010805291606507263001912121645526779536113653020775756*10^-12,
9.33344054962313559794243359121865493436928335383180224238666945820728063025525655836548808428415317933889426550535130382066462031213368885516751465230235566060491142917136569207441645701054679826425697217163901938064900127006472890783962534076014742909945483916619864113434446860386975315278353706626*10^-15,
-1.81952158284214495176005315146531086156653791596180131663435921418721797586252941246481617711630721661789375079721821277846653114423053896463985870450386615652520122516692747064880418921059184037700598516147667108520619602342989030465772045394073118547868277959337098682501296464972966835837922011575*10^-16,
6.55056745969533160814421071628247374352383928596940080617052574955923876311756621608122378105058273927009375031987754479494931189362878369873279452007802905692238259156020329173441251596729735625779351334746939766261966805257054679729183965286634533613807927226092327640181856949276900341117610623655*10^-18,
-3.69074234335865556818212758043669350702656853158038753092151243371810896742466748881552137976231328542847185779988816739307799207492077900830765508192908186997009079033381620257200391100203897877573951839872349453931811908768765328749870274534822228600410878096972194003876375189780453236228760777667*10^-19,
2.92631282440275892025370354255555693756874613876800524426561475470270938018281516993469838505290397763186259552088698663156890935631506958654931199481610656146922529624134515017058653262898132218718293542526142944081267776058903350465168039239934685351339487610448843545659577007326174500062689032630*10^-20,
-3.03715853151891449622151519703998731827848033970384926729211604495579002631059370925988641022371832302162551786392008972624912262488520930944326062623638786025871812993943194359557547196713774033465163302414589776278340656692384541048399755915109531667403663856332740125369895900545146093451964707925*10^-21}

The entry for $k=1$ is clearly bogus. The remaining entries look like they are close but only converging to the right answer for larger $k$. In other words, this appears to be an asymptotic series that got truncated but is nevertheless presented as an equality.

My questions are

  1. Is the functions.wolfram.com entry correct, or is it in error? I ask here first rather than submit a bug report because I struggle to believe such a blatant error would have gone unnoticed since 2001!
  2. If it really is correct as stated, can you please provide an independent reference?
  3. Also if correct, why doesn't Mathematica evaluate to this answer? Mathematica internal software error or user error?
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  • 1
    $\begingroup$ Are you aware that AiryAiZero[] is built-in? $\endgroup$
    – J. M.'s torpor
    Jul 1 at 5:58
  • $\begingroup$ Yes, see the first Table[] evaluation. I am comparing $f(d)$ vs. AiryAiZero[]. $\endgroup$
    – user47363
    Jul 1 at 5:59
  • $\begingroup$ You do realize that your f(d) is an asymptotic expansion, yes? $\endgroup$
    – ciao
    Jul 1 at 6:24
  • 1
    $\begingroup$ See the accepted answer. I suspected it was an asymptotic expansion, but Wolfram presented it as an exact equality, which is the confirmed error. $\endgroup$
    – user47363
    Jul 1 at 6:34
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The canonical reference for asymptotic series for zeroes is Fabijonas and Olver's paper, where the formula for deriving those coefficients is displayed (and see the DLMF as well).

In other words, your observation is spot-on that these are asymptotic approximations for the zeroes, and are usually only expected to be good for large $k$. In that respect, one should be surprised it even works for small $k=2,3,4$, and the behavior for $k=1$ is what's more typical behavior for asymptotic series.

So, do e-mail them at comments(AT)functions(DOT)wolfram(DOT)com informing them that the $=$ should have been a $\sim$ instead.

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  • 1
    $\begingroup$ I will do just that. Thanks for the swift reply with a better reference than my usual Abramowitz & Stegun. $\endgroup$
    – user47363
    Jul 1 at 6:20

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