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.   
      
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