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Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 10 years ago

As of 7:30 AM 11/19/04 it seems that images are not showing up in the messages, They do, however, show up in the Post Preview section. Here is an image:

POSTED BY: Marvin Ray Burns

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Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 10 years ago

I was also having issues with the following post. I had to remove some data before it appeared nice again. Where the upper limit of the following integrals shows Infinity, it is meant to be the (Ultraviolet limit of the sequence) as mentioned by Mathar here: I derived a new formula for computing the integral analog of the MRB constant': f[x_]:=x^(1/x);-((2 I)/\[Pi]^3) + 1/\[Pi]^2 - ( 2 I)/\[Pi] + (I/Pi)^3* Integrate[(-1)^xD[f[x], {x, 3}], {x, 1, Infinity}] In traditional form that is M1= Using it I computed 2000 digits in only 10.8 minutes: In[131]:= Timing[f[x_] = x^(1/x); a = N[1/\[Pi]^2 + (1/Pi)^3 NIntegrate[Sin[Pix]D[f[x], {x, 3}], {x, 1, Infinity}, WorkingPrecision -> 4000], 2000]; b = N[2/\[Pi]^3 + 2/\[Pi] + (1/Pi)^3* NIntegrate[Cos[Pi x]*D[f[x], {x, 3}], {x, 1, Infinity}, WorkingPrecision -> 4000], 2000]; Print[N[Sqrt[a^2 + b^2], 2000]]] During evaluation of In[131]:= 0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447921147157 Out[131]= {653.145, Null} I am presently computing 10,000 digits using that formula. Come back here for results! That formula didn't work out; I will try one of the following formulas. Here are 2 more, more advanced formulas; remember f(x) is x^(1/x): I did finish a 5,000 digit computation using M1= in 48.11 minutes. Here are the 5000 digits:of the magnitude: 0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447529208297180672654123457301218758892278525894167935930983363218877512533994251978272092700003994136520699813263053327399132641690231179063314931546906927612775633995348209911166678724589467821767106592498663827057034363632241807121831546175498178011687284590439293322231263406301066863589072717290630291441982684113819198880100231182613587798104863611185433976009254862585527222843445901958943153561148829083242874018226480554274231391324767376148485531787767908124831873688579979114662856184612164534836370699371440464263768724668291617743681719766849740663590277737977490693183461320266666793472116774276618408124767965369796362732668987556797338128876129264558867657737417548617146808592137056879602982206609613881069490166381528825180204703315896719667069923077454352649723496033985893188309150391579573916059639453655188856334980355047281560296288150836680499821806918067869468571687709518088408966653716009356556714281694904914038988996962213833530636987279769672200413448893419914190954063100962251649102614676944333201213024711868954772741991675045198246947499574872027800654821823797116399297131866662866832215332914761325880983081211272181775518951539503852063119472301382766303820851467743266039356123495461914463960644386394228342211998370152351720235034997434035743513051754761571835043769475528640144621307760159481496713401409374957729200400650100318226988524015127382509490642900236553851499823658269458873976032051355393161653806016080446394196719312454167915154602448638624354575153334932298393406734174580316934939632892851077461038399470015366439910136971186909599331204517462262508377673477745789645309425145559198802530351403897927622891172233239135167420567162398873965477371498335087310395422796362380227536212159184529243644094285328763286873653399867593200891823468738537356817916009007206857590792983184556882143118383332812491747733056313117179696094921120670802012310012864110800437831852620698327457619035904268498030693438632685623213366864129523404256345542376567721287706234359125016588483777876970236084456277023948551334490591022594253744077631232660869593809453087749830900393202787736482133628148979992109544954840067942735030391105496026321872468122542495017023785810605820545392820104069279893067324597299043883381251767370331206913429284614563732308018369972360638019778425246546329838131639355043236388708044857300408692365733932897876809202025693305332974091411983635619038514442263783801745983300121464879550146672827072002317686396598587702487509572349422593441184802476344187280014450860069307120621758277552124841158659386176036703247124389223327008210072318671884895179305778728051888524412158486781863155034447221379906386062559915129172725833420555901857729690605950941678587057025641848365090809750870051863842805803189784976076099574956436664131457150096711473033060684065060747340764998195621425524824611657787212347497307297184843276100338110267863618974154272345482369968216663233417338501929114697679974461999040589290327155974468087040862022522065912789 I'm getting closer to 10K digits of M1: Using , where f(x)=x^(1/x). I got approx. 10K digits of the imaginary part, but the real part was a little garbled.

I was also having issues with the following post. I had to remove some data before it appeared nice again. Where the upper limit of the following integrals shows Infinity, it is meant to be the (Ultraviolet limit of the sequence) as mentioned by Mathar here:

enter image description here

I derived a new formula for computing the integral analog of the MRB constant':

  f[x_]:=x^(1/x);-((2 I)/\[Pi]^3) + 1/\[Pi]^2 - (
  2 I)/\[Pi] + (I/Pi)^3*
  Integrate[(-1)^x*D[f[x], {x, 3}], {x, 1, Infinity}]

In traditional form that is M1= enter image description here

Using it I computed 2000 digits in only 10.8 minutes:

In[131]:= Timing[f[x_] = x^(1/x); 
 a = N[1/\[Pi]^2 + (1/Pi)^3*
     NIntegrate[Sin[Pi*x]*D[f[x], {x, 3}], {x, 1, Infinity}, 
      WorkingPrecision -> 4000], 2000]; 
 b = N[2/\[Pi]^3 + 
    2/\[Pi] + (1/Pi)^3*
     NIntegrate[Cos[Pi x]*D[f[x], {x, 3}], {x, 1, Infinity}, 
      WorkingPrecision -> 4000], 2000]; 
 Print[N[Sqrt[a^2 + b^2], 2000]]]

During evaluation of In[131]:= 0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447921147157

Out[131]= {653.145, Null}

I am presently computing 10,000 digits using that formula. Come back here for results!

That formula didn't work out; I will try one of the following formulas.

Here are 2 more, more advanced formulas; remember f(x) is x^(1/x):

enter image description here

I did finish a 5,000 digit computation using M1=

enter image description here

in 48.11 minutes.

Here are the 5000 digits:of the magnitude:

0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447529208297180672654123457301218758892278525894167935930983363218877512533994251978272092700003994136520699813263053327399132641690231179063314931546906927612775633995348209911166678724589467821767106592498663827057034363632241807121831546175498178011687284590439293322231263406301066863589072717290630291441982684113819198880100231182613587798104863611185433976009254862585527222843445901958943153561148829083242874018226480554274231391324767376148485531787767908124831873688579979114662856184612164534836370699371440464263768724668291617743681719766849740663590277737977490693183461320266666793472116774276618408124767965369796362732668987556797338128876129264558867657737417548617146808592137056879602982206609613881069490166381528825180204703315896719667069923077454352649723496033985893188309150391579573916059639453655188856334980355047281560296288150836680499821806918067869468571687709518088408966653716009356556714281694904914038988996962213833530636987279769672200413448893419914190954063100962251649102614676944333201213024711868954772741991675045198246947499574872027800654821823797116399297131866662866832215332914761325880983081211272181775518951539503852063119472301382766303820851467743266039356123495461914463960644386394228342211998370152351720235034997434035743513051754761571835043769475528640144621307760159481496713401409374957729200400650100318226988524015127382509490642900236553851499823658269458873976032051355393161653806016080446394196719312454167915154602448638624354575153334932298393406734174580316934939632892851077461038399470015366439910136971186909599331204517462262508377673477745789645309425145559198802530351403897927622891172233239135167420567162398873965477371498335087310395422796362380227536212159184529243644094285328763286873653399867593200891823468738537356817916009007206857590792983184556882143118383332812491747733056313117179696094921120670802012310012864110800437831852620698327457619035904268498030693438632685623213366864129523404256345542376567721287706234359125016588483777876970236084456277023948551334490591022594253744077631232660869593809453087749830900393202787736482133628148979992109544954840067942735030391105496026321872468122542495017023785810605820545392820104069279893067324597299043883381251767370331206913429284614563732308018369972360638019778425246546329838131639355043236388708044857300408692365733932897876809202025693305332974091411983635619038514442263783801745983300121464879550146672827072002317686396598587702487509572349422593441184802476344187280014450860069307120621758277552124841158659386176036703247124389223327008210072318671884895179305778728051888524412158486781863155034447221379906386062559915129172725833420555901857729690605950941678587057025641848365090809750870051863842805803189784976076099574956436664131457150096711473033060684065060747340764998195621425524824611657787212347497307297184843276100338110267863618974154272345482369968216663233417338501929114697679974461999040589290327155974468087040862022522065912789

I'm getting closer to 10K digits of M1: Using enter image description here , where f(x)=x^(1/x).

I got approx. 10K digits of the imaginary part, but the real part was a little garbled.

POSTED BY: Marvin Ray Burns

Marvin Ray Burns

Marvin Ray Burns, original investigator of the MRB constant

Posted 10 years ago

I was having trouble formatting the following in the following thread. I had some text placed immediately after an image. Using formula 5 on page 3 of http://arxiv.org/pdf/0912.3844v3.pdf . We can compute a great deal of digits of the integral analog of the MRB constant' (I once called it the MKB constant, named after Marsha Kell-Burns my, now ex, wife.) In the paper Mathar simply calls it M1. This time we will compute the Imaginary part first to at least 500 digits: a[1] = 0; For[n = 1, n < 11, a[n] = N[2/Pi - 1/PiNIntegrate[ Cos[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity}, WorkingPrecision -> 100n], 50 n]; Print[a[n] - a[n - 1]], n++]; Print[a[11]] giving 0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787 0.10^-101 0.10^-151 0.10^-201 0.10^-251 0.10^-301 0.10^-351 0.10^-401 0.10^-451 0.10^-501 0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214786722379645160914886099586780498831455794087390519118879988351918366211827085883779918191195794251385436100844782462528597869421390620796113023053439642582325892202911183326091512210367124716901047132601108752764946385830438156754378694878046808312868541961166205744280461776232345922905313658259576212809654022016030244583148587352474339130505540080799774619683572540292971258866450201101870835703060314349396491402064932644813564545345219868887520120 . Likewise the real part: b[1] = 0; For[n = 1, n < 11, b[n] = N[-1/Pi* NIntegrate[Sin[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity}, WorkingPrecision -> 100n], 50 n]; Print[b[n] - b[n - 1]], n++]; Print[b[11]] giving 0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821 0.10^-102 0.10^-152 0.10^-202 0.10^-252 0.10^-302 0.10^-352 0.10^-402 0.10^-452 0.10^-502 0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723820944073396910971792699904464538475364292258443860652193330471222906120205483985764336623434898438270710499897053952312269178485299032185072743545220051257328105422174249313177670295863771714489658779291185716175115405623656039914848817528200250723061535734571065031458992196831648681239079549382556509741967588147362548743205919028695774572411439927516593391029992733107982746794845130889328251307263102570083031527430861023428334369104098217022622689 . Then the magnitude: N[Sqrt[a[11]^2 + b[11]^2], 500] giving 0.68765236892769436980931240936544016493963738490362254179507101010743\ 3662534784937068627298240498468188731929334335466123286287665409457565\ 9577211580255650416284625143925097120589697986500952590195706813170472\ 5387265069668971286335322245474865156721299946377659227025219748069576\ 0895993932096027520027641920489863095279507385793449828250341732295653\ 3809181101532087948181335825805498812728097520936901677028741356923292\ 2644964771090329726483682930417491673753430878118054062296678424687465\ 624513174205 . That checks with the 200 digits computed by the quadosc command in mpmath by FelisPhasma at https://github.com/FelisPhasma/MKB-Constant . I just now finished 750 digits, (about the max with formula 5 from the paper, as far as Mathematica is concerned). Here is the work: a[1] = 0; For[n = 1, n < 16, a[n] = N[2/Pi - 1/PiNIntegrate[ Cos[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity}, WorkingPrecision -> 100n], 50 n]; Print[a[n] - a[n - 1]], n++]; Print[a[16]]; b[1] = 0; For[n = 1, n < 16, b[n] = N[-1/Pi* NIntegrate[Sin[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity}, WorkingPrecision -> 100n], 50 n]; Print[b[n] - b[n - 1]], n++]; Print[b[16]]; Print[N[Sqrt[a[16]^2 + b[16]^2], 750]] 0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787 0.10^-101 0.10^-151 0.10^-201 0.10^-251 0.10^-301 0.10^-351 0.10^-401 0.10^-451 0.10^-501 0.10^-551 0.10^-601 3.10^-650 -4.10^-700 -2.610^-749 0.68400038943793212918274445999266112671099148265499943432263037713815\ 3058124976638150959834212721478672237964516091488609958678049883145579\ 4087390519118879988351918366211827085883779918191195794251385436100844\ 7824625285978694213906207961130230534396425823258922029111833260915122\ 1036712471690104713260110875276494638583043815675437869487804680831286\ 8541961166205744280461776232345922905313658259576212809654022016030244\ 5831485873524743391305055400807997746196835725402929712588664502011018\ 7083570306031434939649140206493264481356454534521986888752011950353818\ 1776359577265099302389566135475579468144849763261779452665955246258699\ 8679271659049208654746533234375478909962633090080006358213908728990850\ 5026759549928935029206442637425786005036048098598304092996753145589012\ 64547453361707037686708654522699 0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821 0.10^-102 0.10^-152 0.10^-202 0.10^-252 0.10^-302 0.10^-352 0.10^-402 0.10^-452 0.10^-502 2.10^-551 -1.10^-600 1.810^-650 1.2710^-699 4.3410^-749 0.07077603931152880353952802183028200136575469620336302758317278816361\ 8457264382036580831881266177238209440733969109717926999044645384753642\ 9225844386065219333047122290612020548398576433662343489843827071049989\ 7053952312269178485299032185072743545220051257328105422174249313177670\ 2958637717144896587792911857161751154056236560399148488175282002507230\ 6153573457106503145899219683164868123907954938255650974196758814736254\ 8743205919028695774572411439927516593391029992733107982746794845130889\ 3282513072631025700830315274308610234283343691040982170226226904594029\ 7055093272952022662549075225941956559080574835998923469310063614655255\ 0629713179601483134045038416878054929072981851045829413286377842843667\ 5378730394247519728064887287780998671021887797977772522419765594172569\ 277490031071938177749184834961300 0.687652368927694369809312409365440164939637384903622541795071010107433662534784937068627298240498468188731929334335466123286287665409457565957721158025565041628462514392509712058969798650095259019570681317047253872650696689712863353222454748651567212999463776592270252197480695760895993932096027520027641920489863095279507385793449828250341732295653380918110153208794818133582580549881272809752093690167702874135692329226449647710903297264836829304174916737534308781180540622966784246874656245131742049004832216427665542900559350289936114782223424261285828326467186036500189315374147638489679365569122714398706519530651330568884655048857998738535162606116788633540389660052822237449082894798620397228331715198160243676576563833057235963591510865254600 Using formula 7 from the same paper, . (Treating it as we did formula 3), First, the imaginary part to at least 1000 digits:: a[1] = 0; For[n = 1, n < 21, a[n] = N[2/Pi + 1/Pi^2 NIntegrate[ Sin[x Pi] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/ x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n]; Print[a[n] - a[n - 1]], n++]; Print[a[21]] 0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787 0.10^-101 0.10^-151 0.10^-201 0.10^-251 0.10^-301 0.10^-351 0.10^-401 0.10^-451 0.10^-501 0.10^-551 0.10^-601 0.10^-651 0.10^-701 0.10^-751 0.10^-801 0.10^-851 0.10^-901 -2.10^-950 5.10^-1000 0.684000389437932129182744459992661126710991482654999434322630377138153058124976638150959834212721478672237964516091488609958678049883145579408739051911887998835191836621182708588377991819119579425138543610084478246252859786942139062079611302305343964258232589220291118332609151221036712471690104713260110875276494638583043815675437869487804680831286854196116620574428046177623234592290531365825957621280965402201603024458314858735247433913050554008079977461968357254029297125886645020110187083570306031434939649140206493264481356454534521986888752011950353818177635957726509930238956613547557946814484976326177945266595524625869986792716590492086547465332343754789099626330900800063582139087289908505026759549928935029206442637425786005036048098598304092996753145589012645474533617070376867086545228223060940434935219252885333298390272342234952870883304116640409421452765284609364941205344122569781634782508368641126766528707019957340895061936246645065753101916781254557006989818409283317145837167345971516970849116096077030635788389165381066055992688 Then the real part to at least 1000 digits: b[1] = 0; For[n = 1, n < 21, b[n] = N[1/Pi^2 - 1/Pi^2 NIntegrate[ Cos[Pi x] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/ x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n]; Print[b[n] - b[n - 1]], n++]; Print[b[21]] 0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821 0.10^-102 0.10^-152 0.10^-202 0.10^-252 0.10^-302 0.10^-352 0.10^-402 0.10^-452 0.10^-502 0.10^-552 0.10^-602 0.10^-652 0.10^-702 0.10^-752 0.10^-802 0.10^-852 -3.10^-901 8.10^-951 -4.610^-1000 0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382094407339691097179269990446453847536429225844386065219333047122290612020548398576433662343489843827071049989705395231226917848529903218507274354522005125732810542217424931317767029586377171448965877929118571617511540562365603991484881752820025072306153573457106503145899219683164868123907954938255650974196758814736254874320591902869577457241143992751659339102999273310798274679484513088932825130726310257008303152743086102342833436910409821702262269045940297055093272952022662549075225941956559080574835998923469310063614655255062971317960148313404503841687805492907298185104582941328637784284366753787303942475197280648872877809986710218877979777725224197655941725692774900310719381777491848349627938468198411955193898347075098152638657614980900350262780319142430252921925131515239611841070722530473939496294305264627977744876814858325335947117076721493110160508928494597906728688873533031986215124467678736429981544321187124269147141804397293341613 Then the magnitude: In[97]:= N[Sqrt[a[21]^2 + b[21]^2], 1000] Out[97]= 0.\ 6876523689276943698093124093654401649396373849036225417950710101074336\ 6253478493706862729824049846818873192933433546612328628766540945756595\ 7721158025565041628462514392509712058969798650095259019570681317047253\ 8726506966897128633532224547486515672129994637765922702521974806957608\ 9599393209602752002764192048986309527950738579344982825034173229565338\ 0918110153208794818133582580549881272809752093690167702874135692329226\ 4496477109032972648368293041749167375343087811805406229667842468746562\ 4513174204900483221642766554290055935028993611478222342426128582832646\ 7186036500189315374147638489679365569122714398706519530651330568884655\ 0488579987385351626061167886335403896600528222374490828947986203972283\ 3171519816024367657656383305723596359151086525460036387486837632622334\ 2987257095524637683005910353149353985736118868884201748241906260834981\ 7303422370398413326428269921074045506558966667483453656748906071577744\ 4147548424388220133662816274116986724576330176058912438027319979840883\ 05950589130911719199

I was having trouble formatting the following in the following thread. I had some text placed immediately after an image.

Using formula 5 on page 3 of http://arxiv.org/pdf/0912.3844v3.pdf

. enter image description here

We can compute a great deal of digits of the integral analog of the MRB constant' (I once called it the MKB constant, named after Marsha Kell-Burns my, now ex, wife.) In the paper Mathar simply calls it M1.

This time we will compute the Imaginary part first to at least 500 digits:

a[1] = 0; For[n = 1, n < 11, a[n] = N[2/Pi - 1/Pi*NIntegrate[ Cos[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity}, WorkingPrecision -> 100n], 50 n]; Print[a[n] - a[n - 1]], n++]; Print[a[11]]

giving

0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787

0.*10^-101

0.*10^-151

0.*10^-201

0.*10^-251

0.*10^-301

0.*10^-351

0.*10^-401

0.*10^-451

0.*10^-501

0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214786722379645160914886099586780498831455794087390519118879988351918366211827085883779918191195794251385436100844782462528597869421390620796113023053439642582325892202911183326091512210367124716901047132601108752764946385830438156754378694878046808312868541961166205744280461776232345922905313658259576212809654022016030244583148587352474339130505540080799774619683572540292971258866450201101870835703060314349396491402064932644813564545345219868887520120

. Likewise the real part:

b[1] = 0; For[n = 1, n < 11, 
 b[n] = N[-1/Pi*
    NIntegrate[Sin[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity}, 
     WorkingPrecision -> 100*n], 50 n]; Print[b[n] - b[n - 1]], 
 n++]; Print[b[11]]

giving

0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821

0.*10^-102

0.*10^-152

0.*10^-202

0.*10^-252

0.*10^-302

0.*10^-352

0.*10^-402

0.*10^-452

0.*10^-502

0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723820944073396910971792699904464538475364292258443860652193330471222906120205483985764336623434898438270710499897053952312269178485299032185072743545220051257328105422174249313177670295863771714489658779291185716175115405623656039914848817528200250723061535734571065031458992196831648681239079549382556509741967588147362548743205919028695774572411439927516593391029992733107982746794845130889328251307263102570083031527430861023428334369104098217022622689

. Then the magnitude:

N[Sqrt[a[11]^2 + b[11]^2], 500]

giving

0.68765236892769436980931240936544016493963738490362254179507101010743\
3662534784937068627298240498468188731929334335466123286287665409457565\
9577211580255650416284625143925097120589697986500952590195706813170472\
5387265069668971286335322245474865156721299946377659227025219748069576\
0895993932096027520027641920489863095279507385793449828250341732295653\
3809181101532087948181335825805498812728097520936901677028741356923292\
2644964771090329726483682930417491673753430878118054062296678424687465\
624513174205

. That checks with the 200 digits computed by the quadosc command in mpmath by FelisPhasma at https://github.com/FelisPhasma/MKB-Constant .

I just now finished 750 digits, (about the max with formula 5 from the paper, as far as Mathematica is concerned).

Here is the work:

a[1] = 0; For[n = 1, n < 16, 
 a[n] = N[2/Pi - 
    1/Pi*NIntegrate[
      Cos[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity}, 
      WorkingPrecision -> 100*n], 50 n]; Print[a[n] - a[n - 1]], 
 n++]; Print[a[16]]; 
b[1] = 0; For[n = 1, n < 16, 
 b[n] = N[-1/Pi*
    NIntegrate[Sin[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity}, 
     WorkingPrecision -> 100*n], 50 n]; Print[b[n] - b[n - 1]], 
 n++]; Print[b[16]]; Print[N[Sqrt[a[16]^2 + b[16]^2], 750]]

0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787

0.*10^-101

0.*10^-151

0.*10^-201

0.*10^-251

0.*10^-301

0.*10^-351

0.*10^-401

0.*10^-451

0.*10^-501

0.*10^-551

0.*10^-601

3.*10^-650

-4.*10^-700

-2.6*10^-749

0.68400038943793212918274445999266112671099148265499943432263037713815\
3058124976638150959834212721478672237964516091488609958678049883145579\
4087390519118879988351918366211827085883779918191195794251385436100844\
7824625285978694213906207961130230534396425823258922029111833260915122\
1036712471690104713260110875276494638583043815675437869487804680831286\
8541961166205744280461776232345922905313658259576212809654022016030244\
5831485873524743391305055400807997746196835725402929712588664502011018\
7083570306031434939649140206493264481356454534521986888752011950353818\
1776359577265099302389566135475579468144849763261779452665955246258699\
8679271659049208654746533234375478909962633090080006358213908728990850\
5026759549928935029206442637425786005036048098598304092996753145589012\
64547453361707037686708654522699


0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821

0.*10^-102

0.*10^-152

0.*10^-202

0.*10^-252

0.*10^-302

0.*10^-352

0.*10^-402

0.*10^-452

0.*10^-502

2.*10^-551

-1.*10^-600

1.8*10^-650

1.27*10^-699

4.34*10^-749

0.07077603931152880353952802183028200136575469620336302758317278816361\
8457264382036580831881266177238209440733969109717926999044645384753642\
9225844386065219333047122290612020548398576433662343489843827071049989\
7053952312269178485299032185072743545220051257328105422174249313177670\
2958637717144896587792911857161751154056236560399148488175282002507230\
6153573457106503145899219683164868123907954938255650974196758814736254\
8743205919028695774572411439927516593391029992733107982746794845130889\
3282513072631025700830315274308610234283343691040982170226226904594029\
7055093272952022662549075225941956559080574835998923469310063614655255\
0629713179601483134045038416878054929072981851045829413286377842843667\
5378730394247519728064887287780998671021887797977772522419765594172569\
277490031071938177749184834961300

0.687652368927694369809312409365440164939637384903622541795071010107433662534784937068627298240498468188731929334335466123286287665409457565957721158025565041628462514392509712058969798650095259019570681317047253872650696689712863353222454748651567212999463776592270252197480695760895993932096027520027641920489863095279507385793449828250341732295653380918110153208794818133582580549881272809752093690167702874135692329226449647710903297264836829304174916737534308781180540622966784246874656245131742049004832216427665542900559350289936114782223424261285828326467186036500189315374147638489679365569122714398706519530651330568884655048857998738535162606116788633540389660052822237449082894798620397228331715198160243676576563833057235963591510865254600

Using formula 7 from the same paper,

enter image description here .

(Treating it as we did formula 3), First, the imaginary part to at least 1000 digits::

a[1] = 0; For[n = 1, n < 21, 
 a[n] = N[2/Pi + 
    1/Pi^2 NIntegrate[
      Sin[x Pi] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/
        x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n];
 Print[a[n] - a[n - 1]], n++]; Print[a[21]]

0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787

0.*10^-101

0.*10^-151

0.*10^-201

0.*10^-251

0.*10^-301

0.*10^-351

0.*10^-401

0.*10^-451

0.*10^-501

0.*10^-551

0.*10^-601

0.*10^-651

0.*10^-701

0.*10^-751

0.*10^-801

0.*10^-851

0.*10^-901

-2.*10^-950

5.*10^-1000

0.684000389437932129182744459992661126710991482654999434322630377138153058124976638150959834212721478672237964516091488609958678049883145579408739051911887998835191836621182708588377991819119579425138543610084478246252859786942139062079611302305343964258232589220291118332609151221036712471690104713260110875276494638583043815675437869487804680831286854196116620574428046177623234592290531365825957621280965402201603024458314858735247433913050554008079977461968357254029297125886645020110187083570306031434939649140206493264481356454534521986888752011950353818177635957726509930238956613547557946814484976326177945266595524625869986792716590492086547465332343754789099626330900800063582139087289908505026759549928935029206442637425786005036048098598304092996753145589012645474533617070376867086545228223060940434935219252885333298390272342234952870883304116640409421452765284609364941205344122569781634782508368641126766528707019957340895061936246645065753101916781254557006989818409283317145837167345971516970849116096077030635788389165381066055992688

Then the real part to at least 1000 digits:

b[1] = 0; For[n = 1, n < 21, 
 b[n] = N[1/Pi^2 - 
    1/Pi^2 NIntegrate[
      Cos[Pi x] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/
        x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n];
 Print[b[n] - b[n - 1]], n++]; Print[b[21]]

0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821

0.*10^-102

0.*10^-152

0.*10^-202

0.*10^-252

0.*10^-302

0.*10^-352

0.*10^-402

0.*10^-452

0.*10^-502

0.*10^-552

0.*10^-602

0.*10^-652

0.*10^-702

0.*10^-752

0.*10^-802

0.*10^-852

-3.*10^-901

8.*10^-951

-4.6*10^-1000

0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382094407339691097179269990446453847536429225844386065219333047122290612020548398576433662343489843827071049989705395231226917848529903218507274354522005125732810542217424931317767029586377171448965877929118571617511540562365603991484881752820025072306153573457106503145899219683164868123907954938255650974196758814736254874320591902869577457241143992751659339102999273310798274679484513088932825130726310257008303152743086102342833436910409821702262269045940297055093272952022662549075225941956559080574835998923469310063614655255062971317960148313404503841687805492907298185104582941328637784284366753787303942475197280648872877809986710218877979777725224197655941725692774900310719381777491848349627938468198411955193898347075098152638657614980900350262780319142430252921925131515239611841070722530473939496294305264627977744876814858325335947117076721493110160508928494597906728688873533031986215124467678736429981544321187124269147141804397293341613

Then the magnitude:

In[97]:= N[Sqrt[a[21]^2 + b[21]^2], 1000]

Out[97]= 0.\
6876523689276943698093124093654401649396373849036225417950710101074336\
6253478493706862729824049846818873192933433546612328628766540945756595\
7721158025565041628462514392509712058969798650095259019570681317047253\
8726506966897128633532224547486515672129994637765922702521974806957608\
9599393209602752002764192048986309527950738579344982825034173229565338\
0918110153208794818133582580549881272809752093690167702874135692329226\
4496477109032972648368293041749167375343087811805406229667842468746562\
4513174204900483221642766554290055935028993611478222342426128582832646\
7186036500189315374147638489679365569122714398706519530651330568884655\
0488579987385351626061167886335403896600528222374490828947986203972283\
3171519816024367657656383305723596359151086525460036387486837632622334\
2987257095524637683005910353149353985736118868884201748241906260834981\
7303422370398413326428269921074045506558966667483453656748906071577744\
4147548424388220133662816274116986724576330176058912438027319979840883\
05950589130911719199

POSTED BY: Marvin Ray Burns

Bruce Miller

Bruce Miller, Wolfram Research

Posted 10 years ago

Making a guess (feel free to correct me), when you said images are not showing up, did you mean in the upper area, where you type? What goes there for an image is just a reference so the web server can find the image file when it is needed. If your image was not appearing in the saved post, there was a temporary glitch. It appears now.

POSTED BY: Bruce Miller

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