A sum for

This notebook explores the relationship between a truncated sum and an integral involving Meijer G-functions. Here's a summary of the content:

Derivation of
REG
I
:

◼
  • The notebook starts with a truncation of a sum to a finite number of terms, specifically 5 terms.
    • ◼
  • The expression involves the differentiation of a power of a complex expression and evaluates the limit as ε approaches zero from above.
    • ◼
  • The result includes Meijer G-functions, which are special functions that generalize many classical functions.

    • Meijer G-function Scheme:

    ◼
  • A function
    f[n]
    is defined using the
    MeijerG
    function.
    • ◼
  • The notebook demonstrates that the sum of
    
    1-n
    (I/π)
    ·f[n]
    from 1 to
    k
    is very close to an integral involving the exponential

    • Deriving
    REG
    I
    :

    Likewise,

    An example of each identity:

    In[]:=
    mValue=200;(*Choosealargevalueofmforapproximation*)​​integralApprox=NIntegrate[(-1)^nn^(1/n),{n,1,mValueI}]
    Out[]=
    0.070776-0.365691
    In[]:=
    epsilon=10^-15;(*Asmallpositivevalueforε*)​​maxTerms=7;(*Numberoftermsintheseriesforapproximation*)​​N[seriesApprox=Sum[​​(-1)^m/m!D[(epsilon-IPi)^(s-1)Gamma[1-s,epsilon-IPi],{s,m}]/.s->m,​​{m,0,maxTerms}​​],20]
    Out[]=
    0.07077603908225251372-0.36569050489028995872
    In[]:=
    {N[integralApprox-seriesApprox,20]}
    Out[]=
    {2.29268×
    -10
    10
    +1.63615×
    -9
    10
    }
    In[]:=
    (*Truncatesumtoafinitenumberofterms*)​​ε=10^-15;truncatedSum=​​Sum[​​((-1)^m/m!)D[ExpIntegralE[s,-π+ε],{s,m}]/.s->m,​​{m,0,7}(*Exampletruncation*)​​];​​​​(*Numericalevaluation*)​​N[truncatedSum,20]
    Out[]=
    0.07077603908225251372-0.36569050489028995872
    In[]:=
    mValue=200;(*Choosealargevalueofmforapproximation*)​​integralApprox=NIntegrate[(-1)^nn^(1/n),{n,1,mValueI}]
    Out[]=
    0.070776-0.365691
    In[]:=
    N[truncatedSum-integralApprox,20]
    Out[]=
    -2.29268×
    -10
    10
    -1.63615×
    -9
    10
    

    They produce the following MeijerG pattern , so we have the following scheme for
    REG
    I
    :

    In[]:=
    f[n_]:=MeijerG[{{},Table[1,{n+1}]},{{-n+1}~Join~Table[0,{n+1}],{}},-I*Pi]
    The Sum of ((I/Pi)^(1 - n))*f[n] from 1 to k is very close to Integrate[Exp[Pi x I] (Log[x]/x)^k/k!, {x, 1, Infinity I}]
    In[]:=
    Quiet[Table[(NSum[((I/Pi)^(1-n))*f[n],{n,1,k},WorkingPrecision->30]-NIntegrate[Exp[PixI](x^(1/x)-1),{x,1,InfinityI},WorkingPrecision->30]),{k,1,10}]]
    Out[]=
    {-0.013151136322896915896042479590+0.0006897875186108090326933142385,-0.001421490286282695298646021364-0.000513233401500500881066494278,-0.000108221962203470358710102501-0.000083550210242336386452936677,-6.251948266013391446019917×
    -6
    10
    -8.004174116225517844085244×
    -6
    10
    ,-2.81606776161310097239036×
    -7
    10
    -5.72402915924135466272069×
    -7
    10
    ,-9.736326531881161579814×
    -9
    10
    -3.3230120282628222458083×
    -8
    10
    ,-2.29276081718949396455×
    -10
    10
    -1.636148835337933825178×
    -9
    10
    ,-1.022898261731585400×
    -12
    10
    -7.0222569313145840589×
    -11
    10
    ,2.4936746448809275×
    -13
    10
    -2.677645031830936432×
    -12
    10
    ,1.731466464610742×
    -14
    10
    -9.1996602892119102×
    -14
    10
    }
    Does the k’th term of the Sum of ((I/Pi)^(1 - n))*f[n] equal Integrate[Exp[Pi x I] (Log[x]/x)^k/k!, {x, 1, Infinity I}]?
    More precision follows.