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Complex Integral Simplification

Hi Everyone,

I use Mathematica to solve complex integrals that appear in mathematical expressions, for instance, 'coverage probability' and 'average rate' in a given communication network.

I'm stuck at a point where I've tried all possible means but failed to simplify a very important expression, that is attached in 'testIntegral.nb' along with this message. If the expression in the denominator has just the 'z' term without the extra constant of 'x' (that makes the (z + x) term), I easily get a simplified form of the integral in Hypergeometric functions. What I can't understand is, why just adding a constant to the variable of integration 'z' is not letting Mathematica even simplify the expression.

Could you please help me with some pointers or a way in which these integrals could be evaluated using Mathematica?

Hoping for the best.

POSTED BY: Ankit Sharma
Posted 10 years ago

Does transforming the variable of integration from z to w-x work?

IntLap[s_, x_, \[Alpha]_, Lt1_, Lt2_] := 
 Integrate[(1 - 1/(1 + s*(w)^-\[Alpha]))*(w - x), {w, Lt1 + x, 
   Lt2 + x}]
Assuming[s \[Element] Reals && 
  x \[Element] Reals && \[Alpha] \[Element] Reals && 
  Lt1 \[Element] Reals && Lt2 \[Element] Reals && s > 0 && 
  x > 0 && \[Alpha] > 2 && Lt1 > -x && Lt2 > Lt1, 
 Simplify[IntLap[s, x, \[Alpha], Lt1, Lt2]]]

This gives me

-x (-(Lt1 + x) Hypergeometric2F1[1, 1/\[Alpha], 
      1 + 1/\[Alpha], -((Lt1 + x)^\[Alpha]/s)] + (Lt2 + 
       x) Hypergeometric2F1[1, 1/\[Alpha], 
      1 + 1/\[Alpha], -((Lt2 + x)^\[Alpha]/s)]) + (
 s ((Lt1 + x)^(2 - \[Alpha])
      Hypergeometric2F1[1, (-2 + \[Alpha])/\[Alpha], 
      2 - 2/\[Alpha], -s (Lt1 + x)^-\[Alpha]] - (Lt2 + x)^(
     2 - \[Alpha])
      Hypergeometric2F1[1, (-2 + \[Alpha])/\[Alpha], 
      2 - 2/\[Alpha], -s (Lt2 + x)^-\[Alpha]]))/(-2 + \[Alpha])

for output.

POSTED BY: Jim Baldwin
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