0
$\begingroup$

Let me use the notation from Maple http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG for the Meijer G-function. Then let me define,

$f_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$

$f_-(x) = MeijerG( [[-1/2],[]], [[0,0],[]], x )$

Then by numerical evaluation I was able to show that

$\lim_{x\to 0} \frac{f_-(x)}{f_+(x)} = 1/2$

What would the further terms be in an expansion around $x=0$? I guess there will be all sorts of logarithms and other nasty singularities...

(No, Maple 12, which I have installed, is not able to do a series expansion. I don't have access to Mathematica so I don't know about that.)

$\endgroup$

2 Answers 2

2
$\begingroup$

According to Maple 16, both $f_+(x)$ and $f_-(x)$ can be expressed in terms of BesselK:

$$\eqalign{f_+(x) &= \sqrt {\pi }\;{{\rm e}^{x/2}}\;{K_0 \left(x/2\right)}\cr f_-(x) &= \frac{\sqrt {\pi }}{2}{{\rm e}^{x/2}}\left( \left( 1+x \right) { K_0\left(x/2\right)}-x\; { K_1\left(x/2\right)} \right) \cr}$$

We then have the series (not a Taylor series, because of the logarithmic terms)

$$\frac{f_-(x)}{f_+(x)} = {\frac {\ln(x) - 2\;\ln \left( 2 \right) +\gamma+ 2}{2 \; \ln(x) - 4\;\ln \left( 2 \right) +2\; \gamma}}+{\frac {x }{2}}+O \left( {x}^{2} \right) $$

Maple can, of course, provide as many terms as desired, but they look complicated. The coefficient of $x^n$ for odd integers $n > 1$ appears to be $0$, while for even integers $n$ it appears to be of the form $C_n(\ln(x))/(\ln(x) - 2 \ln(2) + \gamma)^{1+n/2}$ where $C_n$ is a polynomial of degree $1+n/2$. Thus if $v = \ln(x)-2 \ln(2) + \gamma$, the series can be written as

$$\eqalign{ &{\frac {v+2}{2v}}+{\frac {x}{2}}+{\frac {2{v}^{2}-2v+1 }{16{v}^{2}}}{x}^{2}-{\frac {8{v}^{3}-20{v}^{2}+ 21v-8}{2048{v}^{3}}}{x}^{4}\cr &+{\frac {72{v}^{4}-276 {v}^{3}+451{v}^{2}-351v+108}{442368{v}^{4}}}{x}^{6}\cr&-{\frac {3168{v}^{5}-16248{v}^{4}+36383\,{v}^{3}-43148 \,{v}^{2}+26784v-6912}{452984832{v}^{5}}}{x}^{8}\cr&+{ \frac {136800{v}^{6}-877560{v}^{5}+2507131{v}^{4}-4011375{v}^{ 3}+3758250{v}^{2}-1944000v+432000}{452984832000{v}^{6}}}{x}^{10}\cr &+O \left( {x}^ {12} \right) \cr} $$

$\endgroup$
1
  • $\begingroup$ It turns out the only reason maple 12 couldn't do the expansion is the I use Fedora 16 linux distribution and there is a known bug in maple that only happens on this platform: mapleprimes.com/questions/… $\endgroup$
    – Daniel
    Oct 26, 2012 at 21:03
2
$\begingroup$

You have access to Mathematica through wolfram alpha.

http://wolframalpha.com

series x=0 of MeijerG[{{-1/2},{}},{{0,0},{}},x] / MeijerG[{{1/2},{}},{{0,0},{}},x]

$\endgroup$
1
  • $\begingroup$ Great, thanks a lot, I always forget wolframalpha! $\endgroup$
    – Daniel
    Oct 26, 2012 at 20:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.