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f(x,u)=0∫uIm(x1+It(1+It))csch(πt)t=CMRB
f(x,u)=Im()csch(πt)t=CMRB
0
∫
u
x
1+It
(1+It)
Besides x=1, u=∞ what parameters in this form can give the same sum?
What parameters for u and x in the form Integrate[
Csch[Pi t] Im[((1 + I t)^(x/(1 + I t)))], {t, 0, u}] makes it equal the MRB constant?
Csch[Pi t] Im[((1 + I t)^(x/(1 + I t)))], {t, 0, u}] makes it equal the MRB constant?
What parameters in that form that can give MRB constant sum?
In[]:=
m=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision100]
Out[]=
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278
In[]:=
Quiet[N[x]/.FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{x,2},WorkingPrecision100]]
Out[]=
0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999572940511
In[]:=
Quiet[x/.FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{x,25},WorkingPrecision100]]
Out[]=
25.65665403510586285599072933607445153794770546058072048626118194900973217186212880099440071247392077
In[]:=
x=25.6566540351058628559907293360744515379477054605807204862611819490097321718621288009944007124739159792146480733342667`100.
Out[]=
25.65665403510586285599072933607445153794770546058072048626118194900973217186212880099440071247391598
In[]:=
m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"]
Out[]=
-9.3470×
-94
10
In[]:=
(results=Table[Quiet[FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision100,AccuracyGoal100,Method"Trapezoidal"],{u,a},WorkingPrecision100]],{a,-3,4}])//TableForm
Out[]//TableForm=
u-3.205281240093347156628042917392342206845495258334899739232149151500845258217206440409918401888147990 |
u-1.975955817063408761652299553542124207955844914033641475935228264550492865978094627264109660184110813 |
u-1.028853359952178482391753039155168552490590931630510790413956782402957713372878008212396966920395894 |
u0.02332059641642379960870201829771316178976827597634872153712935185694505884968182818940208455450915972 |
u1.028851065679287940491239061962065335882572934488973325479147532561027453451485653303655759243419291 |
u1.975930036556044011032057974393627818460615746707424999683678154612112980778643821981630632223625260 |
u3.377688794565491686010340135121834601964847740798723565007400061225550945863584207850080547919162076 |
u4.218664066279720330451890569753298644162048163522884025276106357736553673659421134883269758395781650 |
In[]:=
Quiet[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision->100,AccuracyGoal->100,Method->"Trapezoidal"]/.results]//TableForm
Out[]//TableForm=
-1.54387828979211679160225856368628698239013559× -54 10 |
-1.650587080052960541099285654572141729212307× -56 10 |
-4.74733569309890585575136369263660582070431× -57 10 |
3.88432825092307283199029322311271029659117× -57 10 |
4.72926388624601586314520015635582296283049× -57 10 |
1.653990003627211153185296844342954430897517× -56 10 |
3.387517822661408436528755032035501466434285285× -53 10 |
-1.876525309243100801023411326326237599736649923× -53 10 |
In[]:=
(results2=Table[Quiet[FindRoot[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision60,AccuracyGoal60,Method"Trapezoidal"],{u,a},WorkingPrecision50]],{a,-10,10}])//TableForm
Out[]//TableForm=
u-365.22295293936251072152626085793204637057503999271 |
u-365.22295293936251072152626085793204637057503999270 |
u-365.22295293936251072152626085793204637057503999268 |
u-967.00000000000000000000000000000000000000000000021 |
u-846.00000000000000000000000000000000000000000000018 |
u-725.00000000000000000000000000000000000000000000016 |
u23.982026182324853803608889339082042631551349168612 |
u-3.2052812400933471566280429173762898207215999072394 |
u-1.9759558170634087616522995535419275599858039312417 |
u-1.0288533599521784823917530391551694728347101244931 |
u0.023320596416423799608702018296793701607309992216887 |
u1.0288510656792879404912390619620662542309869785527 |
u1.9759300365560440110320579743934304945391407822302 |
u3.3776887945654916860103401351177406575345223509622 |
u4.2186640662797203304518905718251923173711453766859 |
u4.2186640662797203304518905722660194292848065273921 |
u4.2186640662797203304518905722604776695949964755325 |
u4.2186640662797203304518905527760297705720364891035 |
u8.0000000000000000000000000000000000000000000000015 |
u9.0000000000000000000000000000000000000000000000017 |
u10.000000000000000000000000000000000000000000000002 |
In[]:=
Quiet[m-NIntegrate[Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,u},WorkingPrecision60,AccuracyGoal60,Method"Trapezoidal"]/.results2]//TableForm
Out[]//TableForm=
0.18785964246206712024851793405427323005590309490013878617200368408947723158838742675456267622902370 |
0.18785964246206712024851793405427323005590309490013878617200368408947723158838742675456267622902370 |
0.18785964246206712024851793405427323005590309490013878617200368408947723158838742675456267622902370 |
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278 |
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278 |
0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443217278 |
0.023908155645042418980607687507586137383048607405980473430760 |
-7.3222464811394622609093750277086× -29 10 |
2.865456171810368209714118570696642× -27 10 |
-1.50721278328062721702783292532294× -28 10 |
7.204654170253720194421355231834× -30 10 |
1.50390353522531254043430219385813× -28 10 |
-2.876202653591667468213473413486786× -27 10 |
2.995731153489302635123737606497× -30 10 |
4.1544271865246141799112426746474× -29 10 |
5.0383451992934179436894364723518× -29 10 |
5.0272332211961666139955761714807× -29 10 |
-3.40417214808795011039125231044866× -28 10 |
0.007975358500381911621715400941796461044643239907311225166612 |
0.008972277807109675646259848002801343507542950982931871798894 |
0.009969197541957246074447606339066561351957518154300313131775 |