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I define a piecewise function as

g[x_] := 
  Piecewise[
   {{lambda * Exp[lambda *(x - nBar)]/(1 - Exp[lambda*(nUnder - nBar)]), 
     nUnder <= x <= nBar}}, 0]

I would like to define and compute another function g2, assuming that 0 < nUnder < nBar:

$$g_2(x) = \int g(s-t)g(t)dt$$

g2[s_] := Assuming[{nBar - nUnder > 0}, Integrate[g[s - x]*g[x], {x, nUnder, nBar}]]
g2[s]

The output is

func

or 

Piecewise[
  {{-((E^(lambda*s)*lambda^2*(-2*nBar + s))/(E^(lambda*nBar) - E^(lambda*nUnder))^2), 
    nBar - nUnder > 0 && nBar + nUnder - s < 0 && 2*nBar - s > 0}, 
   {(E^(lambda*s)*lambda^2*(-2*nUnder + s))/(E^(lambda*nBar) - E^(lambda*nUnder))^2, 
    (nBar + nUnder - s == 0 && nBar - nUnder > 0) || 
    (nBar + nUnder - s >= 0 && nBar - nUnder > 0 && 
    2*nUnder - s < 0)}}, 
  0]

in input form.

As you can see, it keeps iterating the assumption nBar - nUnder > 0 — but I thought I enforced that by using Assuming?

  1. How can I simplify g2 to permanently work under the assumption nBar > nUnder > 0? Simplify was of no help.
  2. Similarly, it would help to redefine g2 for the domain 2 nUnder <= s <= 2 nBar, to get rid of some other of the conditions. Can I achieve that somehow?
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  • $\begingroup$ I can't post the question with latex - the system keeps telling me that there is unformatted code in the question... $\endgroup$
    – FooBar
    Jul 1, 2017 at 16:40
  • $\begingroup$ I've replaced the TeX with an image. Roll back if you prefer. $\endgroup$
    – Michael E2
    Jul 2, 2017 at 0:10
  • $\begingroup$ @MichaelE2. think it's my image of the outpu that replaced it :-) $\endgroup$
    – m_goldberg
    Jul 2, 2017 at 0:11
  • $\begingroup$ @m_goldberg The edit history says you replaced my image with yours -- simultaneous editing. :) $\endgroup$
    – Michael E2
    Jul 2, 2017 at 0:13

2 Answers 2

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Include Simplify in definition of g2

g[x_] := Piecewise[{{lambda*
     Exp[lambda*(x - nBar)]/(1 - Exp[lambda*(nUnder - nBar)]), 
    nUnder <= x <= nBar}}, 0]

g2[s_] = Assuming[{nBar > nUnder > 0},
  Integrate[g[s - x]*g[x], {x, nUnder, nBar}] //
   Simplify]

enter image description here

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Try removing the Piecewise definition and only integrate over the required region.

g2[x_] := lambda*Exp[lambda*(x - nBar)]/(1 - Exp[lambda*(nUnder - nBar)])

Integrate[g2[x], {x, nUnder, nBar}]

Integrate[g2[s - x]*g2[x], {x, Max[nUnder, s - nUnder], Min[s - nBar, nbar]}]
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3
  • $\begingroup$ Integrate[g2[s - x]*g2[x], {x, nUnder, nBar}] ensures that g2[x] is only integrated over the required domain, but g2[s-x] in general is not necessarily inside the required domain. $\endgroup$
    – FooBar
    Jul 1, 2017 at 18:59
  • $\begingroup$ So you need to integrate only when both x and s-x are between nUnder and nBar? $\endgroup$
    – John McGee
    Jul 1, 2017 at 19:30
  • $\begingroup$ Yes, that is the case. $\endgroup$
    – FooBar
    Jul 1, 2017 at 20:10

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