Misunderstanding with MeijerG function

The standard way to express MeijerG in terms of more specific special functions is with FunctionExpand. For example

MeijerG[{{1}, {}}, {{1/2, 1, 3/2}, {}}, z]

MeijerG[{{1}, {}}, {{1/2, 1, 3/2}, {}}, z]

FunctionExpand[%]

-2 π z - π^2 z BesselY[1, 2 Sqrt[z]] + π^2 z StruveH[1, 2 Sqrt[z]]

However your expression does not simplify with this approach

FunctionExpand[1/2 π MeijerG[{{}, {}}, {{0, 1/2, 1/2}, {0}}, r^4/256], r > 0]

1/2 π MeijerG[{{}, {}}, {{0, 1/2, 1/2}, {0}}, r^4/256]

The best work around I could find is to express the 3-arg form of MeijerG into a 4-arg form, which will be correct on the positive real axis.

mei = 1/2 π MeijerG[{{}, {}}, {{0, 1/2, 1/2}, {0}}, r^4/256];

FunctionExpand[mei /. {
  MeijerG[as_, bs_, c_. r^n_?Positive] :> MeijerG[as, bs, c^(1/n) r, 1/n]
}]

-2 π KelvinKei[0, r]

Another (less successful) workaround is to solve the ODE your MeijerG satisfies.

DifferentialRootReduce[1/2 π MeijerG[{{}, {}}, {{0, 1/2, 1/2}, {0}}, r^4/256], r]

enter image description here

Now FunctionExpand knows how to handle DifferentialRoot objects, but it seems to spin forever on this input presumably because the solution from DSolve on this ODE has very complicated coefficients.

If we approximate the initial conditions, we can see your expression is really $-2\pi \text{kei}_0(r)$.

DSolveValue[{
  r^3 y[r] + y'[r] - r y''[r] + 2 r^2 y'''[r] + r^3 y''''[r] == 0, 
  y[1] == 3.1101430273071208511784026627334110346453670869257748826627`30., 
  y'[1] == -2.2140054621659709246306930546689437640742733642857570978703`30., 
  y''[1] == 0.412577224007051831900819509950661362815906739014512338784`30., 
  y'''[1] == 1.7377422767006054496810801840889187820973078345984715247552`30.
}, y[r], r] // Chop

-6.28318530717958647692528677 KelvinKei[0, r]

Addressing your second question, my guess is this particular MeijerG isn't evaluating at the origin because it's a (rather complicated looking) branch point:

enter image description here


In addition to Chip's nice answer, another way would be to do a "round trip" using the Mellin transform and its inverse:

InverseMellinTransform[
       MellinTransform[π/2 MeijerG[{{}, {}}, {{0, 1/2, 1/2}, {0}}, r^4/256], r, t], t, r]
   -2 π KelvinKei[0, r]

Bonus: the original integral can in fact be expressed as an appropriate Mellin convolution:

2 π MellinConvolve[BesselJ[0, k], k^2/(1 + k^4), k, r]

Unfortunately, this also returns the normal $G$ function (as with Integrate[]) instead of the more useful (at least in this case) extended $G$ function.

Tags:

Special Functions

Calculus And Analysis

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