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Let {x},{y}>0 and {a},{b}>0. can the function log(1+ax+by) be expressed as a Meijer-G function?

fatimaaa
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  • Please look up MeijerGReduce as suggested, and also Integrate. Make an attempt at writing your expression in Mathematica. I am pretty sure that with this effort, you will have a few people at this site giving valuable help! –  Mar 17 '19 at 20:16
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    Have you looked at MeijerGReduce? – mikado Mar 17 '19 at 18:41
  • Yes. Please look here: http://mathworld.wolfram.com/MeijerG-Function.html. In the first 'special case' listed, let $z=ax+by$. –  Mar 17 '19 at 18:43
  • Unfortunately, I have not worked with Mathematica so far. Can not be converted using MATLAB? – fatimaaa Mar 17 '19 at 18:47
  • MATLAB questions are not on-topic for this site. – J. M. ain't a mathematician Mar 17 '19 at 18:54
  • Actually, i want to solve $\int\limits_{0}^{\infty }{\int\limits_{0}^{\infty }{\ln (1+{{a}{1}}x+{{a}{2}}y)G_{1,1}^{1,1}\left( {{a}{2}}x\left| \begin{matrix} -{{a}{3}} \ {{a}{4}}-1 \ \end{matrix} \right. \right)G{1,1}^{1,1}\left( {{b}{2}}y\left| \begin{matrix} -{{b}{3}} \ {{b}_{4}}-1 \ \end{matrix} \right. \right)}}dxdy$

    as a close-form expression.

     – fatimaaa Mar 17 '19 at 19:02
  • I use mjw's comment and I can rewrite it as $\int\limits_{0}^{\infty }{\int\limits_{0}^{\infty }{G_{2,2}^{1,2}\left( {{a}{1}}x+{{b}{1}}y\left| \begin{matrix} 1,1 \ 1,0 \ \end{matrix} \right. \right)G_{1,1}^{1,1}\left( {{a}{2}}x\left| \begin{matrix} -{{a}{3}} \ {{a}{4}}-1 \ \end{matrix} \right. \right)G{1,1}^{1,1}\left( {{b}{2}}y\left| \begin{matrix} -{{b}{3}} \ {{b}_{4}}-1 \ \end{matrix} \right. \right)}}dxdy$ Is there any solution for this integral? – fatimaaa Mar 17 '19 at 19:05
  • There is a MATLAB version of the Meijer-G function: https://www.mathworks.com/help/symbolic/meijerg.html. –  Mar 17 '19 at 19:50
  • Thank you very much for your valuable answers. Have you got an idea to solve the integral above? – fatimaaa Mar 17 '19 at 20:04
  • f = ...; Integrate[f, {x, 0, \[Infinity]}, {y, 0, \[Infinity]}]; –  Mar 17 '19 at 20:19
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    Why did you not ask about that integral to begin with in your question? Edit your post to include this. – J. M. ain't a mathematician Mar 17 '19 at 20:49

1 Answers1

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This is meant to be an outline of an answer. The details of arguments for the Meijer-G functions still need to be filled in:

z[x_, y_] = a1 x + b2 y;
\[Alpha] = Evaluate@MeijerGReduce[Log[1 + z[x, y]], z[x, y]];
\[Beta] = MeijerG[{{}, {}}, {{}, {}}, a2 x];
\[Gamma] = MeijerG[{{}, {}}, {{}, {}}, b2 y];
f = \[Alpha] \[Beta] \[Gamma];
Integrate[f, {x, 0, \[Infinity]}, {y, 0, \[Infinity]}]
mjw
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