From: edgar@math.ohio-state.edu.invalid   
      
   In article <524FF9CF.6F7CD843@freenet.de>,    
   wrote:   
      
   > Axel Vogt schrieb:   
   > >   
   > > The book says: Int( exp(-x*a) * BesselI(0, b*sqrt(-x^2+1)), x = -1 .. 1) =   
   > > = 2*sinh(a^2+b^2)/sqrt(a^2+b^2)   
   > >   
   > > For a=0 I obtain that it is 2*sinh(b)/b, using Maple (sketch: write Bessel   
   > > as series and integrate termwise; evaluating the infinite series gives it).   
   > > Numerical tests do confirm that.   
   > >   
   > > Is somebody aware of the root of the proof in G&R (can not find an erratum,   
   > > but some usage of the formula) to eliminate the possible typo?   
   >   
   > There is no entry 6.616.5 for   
   >   
   >   INT(exp(-x*a)*BesselI(0, b*sqrt(1-x^2)), x, -1, 1)   
   >   
   > in my 1981 edition of Gradshteyn & Ryzhik; it must have been added   
   > later. I suggest to try the modified evaluation   
   >   
   >   2*sinh(sqrt(a^2+b^2))/sqrt(a^2+b^2)   
   >   
   > which holds at b=0 as well as a=0, since BesselI(0, 0) = 1. It is also   
   > in full agreement with a numerical evaluation of the integral on Derive   
   > giving 2.199018052 for a = 0.3 and b = 0.7. Doing the general integral   
   > symbolically may be difficult; related integrals under 6.616 in my   
   > edition are all taken from Magnus and Oberhettinger, 1948.   
   >   
   > You could file an error report with the G&R editors (after checking the   
   > latest edition).   
   >   
   > Martin.   
      
   In my 1980 copy of G&R (English translation of the fourth Russian   
   edition), there is such an entry, with note 3 meaning "added in the   
   third edition".  However (as Axel suggests) the value is   
   2*sinh(sqrt(a^2+b^2))/sqrt(a^2+b^2)   
      
   --   
   G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/   
      
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