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Making a function out of repeated hyperbola integrations?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122061] Making a function out of repeated hyperbola integrations?
  • From: Nathan McKenzie <kenzidelx at gmail.com>
  • Date: Tue, 11 Oct 2011 04:24:30 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I'm working with the following repeated integrals.  Is there any way
to automate what I'm doing here?

I start with this:

Integrate[ n/x - a, {x, a, n/a}]

The result of that (after a bit of text editing) is a^2 - n Log[a] + n
(-1 + Log[n/a]). That's the first result I want to work with.  Then, I
currently manually edit that result by swapping out n with (n/x), and
have

Integrate[ a^2 - (n/x) Log[a] + (n/x) (-1 + Log[(n/x)/a]), {x, a, n/
a}]

And that resolves to -a^3 + n Log[a] + n Log[a]^2 + 1/2 n Log[n/a^2]^2
+  n (a - (1 + Log[a]) Log[n/a]).  Which is the second result I want
to work with.  The, I manually swap out n with (n/x) and integrate
again on {x, a, n/a}, and repeat this process ad nauseum.  It doesn't
take long before the number of n's for me to edit becomes really
unwieldy and error prone... and ideally I would like to do this many
times in a row (say, up 30 or 40) and have the results around for
random use in other contexts.

What I would really like to be able to do is just have some sort of
function where I can type F[n,a,s], where s is the number of times
integration is performed, and then n and a (which will be actual
numbers) will get evaluated.  I feel like the step where I swap out n
with (n/x) points at something problematic, though.  Is there any way
in Mathematica for me to construct such a function and get my results
automatically?



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