       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:=
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:=
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:=
(Limit[f[x, 1], x -> #1] & ) /@ {0, Infinity}

Out=
{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:=
g[y_] = f[x, m] /. x -> y*m

Out=
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:=
(Simplify[#1, y > 0] & )[D[g[y], y]]

Out=
(4*BesselK[1, y])/y

Take now the derivative of the numerator of the last expression.

In:=
Expand[4*FullSimplify[D[BesselK[1, y], y]]]

Out=
-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:=
oo = FullSimplify[-4*(Integrate[BesselK[0, y], y] + BesselK[1, y])]

Out=
-2*(Pi*y*BesselK[0, y]*StruveL[-1, y] + BesselK[1, y]*(2 +
Pi*y*StruveL[0, y]))

with

In:=
Plot[oo, {y, 0, 10}]

In:=
(Limit[oo, y -> #1] & ) /@ {0, Infinity}

Out=
{-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:=
mei = FullSimplify[oo /. y -> x/m]

Out=
-((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. 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, compltely belongs to him!

=CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> I have the following fuction
>
> In:=
> 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

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