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Re: Question about Integration and citation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126322] Re: Question about Integration and citation
  • From: danl at wolfram.com
  • Date: Tue, 1 May 2012 05:21:45 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jngd3i$la$1@smc.vnet.net>

On Saturday, April 28, 2012 4:26:10 AM UTC-5, mehdimolu wrote:
> Hi,
> I need to perform an integration as follows:
> 
> ---> Integrate[Log(1+a x)*x^n*E^(-b x),{x,0,Infinity}]
> 
> The Mathematica installed on my mac performs the integration and outputs
> some results but when I use wolframalpha,the answer is "No more results
> available", even after pressing "Try again with more time" button.
> 
> the other question is that I need a citation when I use the mentioned
> integration in a paper but I searched in several integration table books and
> http://functions.wolfram.com website but could not find anything for
> citation. 
> can any one, please, help me... 
> -- 
> View this message in context: http://old.nabble.com/Question-about-Integration-and-citation-tp33760346p33760346.html
> Sent from the MathGroup mailing list archive at Nabble.com.

The URL from Murray Eisenberg gives appropriate information for citing the program Mathematica. in addition you could cite the following for the following formulas that rewrite the log and exponential factors as MeijerG functions.

http://functions.wolfram.com/ElementaryFunctions/Exp/26/02/01/0001/

and

http : // functions.wolfram.com/ElementaryFunctions/Log/26/02/01/0002/

I show below the Mathematica evaluations of the appropriate MeijerG functions, to corroborate these conversions.

In[55]:= MeijerG[{{}, {}}, {{0}, {}}, b*x]

Out[55]= E^((-b)*x)

In[56]:= MeijerG[{{1, 1}, {}}, {{1}, {0}}, a*x]

Out[56]= Log[1 + a*x]

To go from these to the integral requires a result involving convolution of two MeijerG functions times a power of x. For this you could cite formulas (21) and (22) from

V. Adamchik and O. Marichev. The algorithm for calculating integrals of hypergeometric type functions and its realization in Reduce system. Proceedings of the 1990 International Symposium on Symbolic and Algebraic Computation p. 212-224. 1990. ACM Press.

Or maybe use reference 19 from that article.

Here is a link to a pdf of the article (might require ACM membership for free access though).

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCUQFjAA&url=http%3A%2F%2Fdl.acm.org%2Fft_gateway.cfm%3Fid%3D96930%26type%3Dpdf&ei=BN-eT-rDJqP30gHRpc3oDg&usg=AFQjCNF2hS-CVX4lsEzKwCgBB4yXahWKhw

Daniel Lichtblau
Wolfram Research



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