Re: MeijerG function

*To*: mathgroup at smc.vnet.net*Subject*: [mg75413] Re: MeijerG function*From*: dimitris <dimmechan at yahoo.com>*Date*: Sun, 29 Apr 2007 03:09:09 -0400 (EDT)*References*: <f0sfk8$n8r$1@smc.vnet.net>

At last I succeeded! Firstly, let In[14]:= f[o_, m_] := MeijerG[{{1}, {}}, {{-2^(-1), 1/2}, {0}}, o^2/(4*m^2)] For m=1 here is a plot of this function In[17]:= Plot[{f[x, 1], -(1/x + 2*Pi)}, {x, 0, 10}, Axes -> False, Frame -> True, PlotStyle -> {Red, Blue}] Oberve that, (it seems...) the function behaves as -1/x as x->0+, and tends to -2*Pi at infinity. However, Mathematica fails to provide these limits. In[18]:= (Limit[f[x, 1], x -> #1] & ) /@ {0, Infinity} Out[18]= {Limit[MeijerG[{{1}, {}}, {{-(1/2), 1/2}, {0}}, x^2/4], x -> 0], Limit[MeijerG[{{1}, {}}, {{-(1/2), 1/2}, {0}}, x^2/4], x -> Infinity]} Now, set In[26]:= g[y_] = f[x, m] /. x -> y*m Out[26]= MeijerG[{{1}, {}}, {{-(1/2), 1/2}, {0}}, y^2/4] Without loss of generality, assume y>0 (y is real) and take the derivative of g[y]. In[27]:= (Simplify[#1, y > 0] & )[D[g[y], y]] Out[27]= (4*BesselK[1, y])/y Take now the derivative of the numerator of the last expression. In[30]:= Expand[4*FullSimplify[D[BesselK[1, y], y]]] Out[30]= -4*BesselK[0, y] - (4*BesselK[1, y])/y With this procedure, someone is able to see that the derivative of g[y] (ie the requested (normalized) MeijerG function), which is equal to (4*BesselK[1, y])/y, can be written as -4*(BesselK[0, y]+D[BesselK[1, y], y]. Hence the function g[y] will be given by the indefinite integral of the last expression. In[35]:= oo = FullSimplify[-4*(Integrate[BesselK[0, y], y] + BesselK[1, y])] Out[35]= -2*(Pi*y*BesselK[0, y]*StruveL[-1, y] + BesselK[1, y]*(2 + Pi*y*StruveL[0, y])) with In[36]:= Plot[oo, {y, 0, 10}] In[37]:= (Limit[oo, y -> #1] & ) /@ {0, Infinity} Out[37]= {-Infinity, -2*Pi} and returning back to the original variable x=y/m, we finally get the following simplified expression for the MeijerG function In[44]:= mei = FullSimplify[oo /. y -> x/m] Out[44]= -((2*(Pi*x*BesselK[0, x/m]*StruveL[-1, x/m] + BesselK[1, x/m]*(2*m + Pi*x*StruveL[0, x/m])))/m) Dimitris PS A collegue of mine (Panagiotis Gourgiotis) came across this MeijerG function during the course of the inversion of one Fourier Transform. He suceeded in writing this function as in Output[44]. He didn't give me more details. He put me as a chalenge if I could use a CAS in order to arrive at his formula (he checked his formula numerically). However, having got "stick" I pushed him in order to give one or two hints. The step appeared at In[30], compltely belongs to him! =CF/=C7 dimitris =DD=E3=F1=E1=F8=E5: > I have the following fuction > > In[11]:= > f = MeijerG[{{1}, {}}, {{-2^(-1), 1/2}, {0}}, o^2/(4*m^2)]; > > where o, m are both reals. > > Can we write down f in terms of other special functions? > > Thanks > Dimitris