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Q: Dickman function

  • To: mathgroup at
  • Subject: [mg20909] Q: Dickman function
  • From: fgrieu at (Francois Grieu)
  • Date: Sun, 21 Nov 1999 15:12:52 -0500 (EST)
  • Sender: owner-wri-mathgroup at

I'm trying to build the Dickman function, defined by

F(x)=1  for x>=1     F(x)=1-|   F(t/(1-t))/t dt   for 0<=x<=1

(ref: Knuth's TAOCP, vol 2, 4.5.4, p383 in third edition)

After a lot of trial and error, I came up with a tentative
minimal definition in Mathematica:


This does works for x>=1/3 both numericaly and symbolicaly.
However both F[0.3] and F[1/4] give an imaginary component, which
is an 'obvious' nonsense; it's numerical value is vanishingly
small, but the expression of F[x,3] is so complex that even
FullSimplify won't cut it. And it's sooo long when x gets small.

More trial and error leads to


Extending the usable range to maybe x>=0.15, but I am afraid I won't
ever get F[0.1] that way.

Any clues ?

  Francois Grieu

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