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Mathematics Calculus Wolfram Language

Unexpected output from Integrate

Carlo Wood

Posted 9 months ago

Hello, I am desperately trying to evaluate the following integral for several days now: Integral[(1/(1+Exp[-x]))Log[1/(1+Exp[-(xw+b)])],{x,-Infinity,Infinity}] I can't get Mathematica to do it; so I tried - at some point - this: Integrate[(1/(1 + Exp[-x]))*Log[1/(1 + Exp[-x])], x] That results in a formula, but it doesn't seem to make any sense?! It returns: https://gyazo.com/8ebda5752271450bbcf73067974f84a8 (I can't get Mathematica to copy anything to clipboard) This contains `Log[-Exp[x]]` which can't exist since Exp[] is always positive?! Can someone please help, Carlo Wood

POSTED BY: Carlo Wood

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 9 months ago

The output contains functions with complex values, but the end result is real: int = Integrate[(1/(1 + Exp[-x]))*Log[1/(1 + Exp[-x])], x] int2 = FullSimplify[ComplexExpand[int], Element[x, Reals]] FullSimplify[ComplexExpand[int] == int2, Element[x, Reals]] Plot[int - int2, {x, -1, 1}] Unfortunately we cannot check the derivative of `int2`, because `int2` contains `Re`, which is not differentiable in the complex sense.

POSTED BY: Gianluca Gorni

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 9 months ago

The imaginary parts will cancel. ii = Integrate[(1/(1 + Exp[-x]))Log[1/(1 + Exp[-x])], x] ( Out[359]=Log[-E^x] Log[1/(1 + E^x)] + 1/2 Log[1/(1 + E^x)]^2 - Log[1/(1 + E^x)] Log[E^x/(1 + E^x)] - PolyLog[2, 1 + E^x] ) hi = Limit[ii, x -> Infinity] lo = Limit[ii, x -> -Infinity] ( Out[360]= -(\[Pi]^2/3) Out[361]= -(\[Pi]^2/6) ) hi - lo ( Out[362]= -(\[Pi]^2/6) *)

The imaginary parts will cancel.

ii = Integrate[(1/(1 + Exp[-x]))*Log[1/(1 + Exp[-x])], x]

(* Out[359]=Log[-E^x] Log[1/(1 + E^x)] + 1/2 Log[1/(1 + E^x)]^2 - 
 Log[1/(1 + E^x)] Log[E^x/(1 + E^x)] - PolyLog[2, 1 + E^x] *)

hi = Limit[ii, x -> Infinity]
lo = Limit[ii, x -> -Infinity]

(* Out[360]= -(\[Pi]^2/3)

Out[361]= -(\[Pi]^2/6) *)

hi - lo

(* Out[362]= -(\[Pi]^2/6) *)

POSTED BY: Daniel Lichtblau

Carlo Wood

Posted 9 months ago

Sorry, I was typing that from my head - I use Integrate in the notebook example. What can you say about the log(-exp(x)) in the answer? I knew that the integral with limits for the simpler form is -pi^2/6; I either need the definite integral with the w and b, or an explanation of the logarithm of a negative value in an answer that should be just real function.

POSTED BY: Carlo Wood

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 9 months ago

Use `Integrate`, not `Integral`. In[316]:= Integrate[(1/(1 + Exp[-x]))* Log[1/(1 + Exp[-x])], {x, -Infinity, Infinity}] Out[316]= -(\[Pi]^2/6) In[317]:= NIntegrate[(1/(1 + Exp[-x]))* Log[1/(1 + Exp[-x])], {x, -Infinity, Infinity}] Out[317]= -1.64493

Use Integrate, not Integral.

In[316]:= Integrate[(1/(1 + Exp[-x]))*
  Log[1/(1 + Exp[-x])], {x, -Infinity, Infinity}]

Out[316]= -(\[Pi]^2/6)

In[317]:= NIntegrate[(1/(1 + Exp[-x]))*
  Log[1/(1 + Exp[-x])], {x, -Infinity, Infinity}]

Out[317]= -1.64493

POSTED BY: Daniel Lichtblau

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