Can't obtain a closed form expression for an integral

Question

I am trying to find the result of quite a complicated double integral. While the first part (being a Principal value integral) gave no particular problems, the second one is not providing any closed expression. Here is my code:

f[x_]:=(p^2 + k^2 - 2 p k x)/(x - (p^2 + k^2 + 1 - ((p^2 - k^2)^2)/4)/(2 p k))
Integrate[f[x], {x, -1, 1}, PrincipalValue -> True]

ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) ArcCoth[(
 8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)), 
k^4 + p^4 < 
  4 + 4 p^2 + 2 k^2 (2 + p^2) && (k - p)^2 (-2 + k + p) (2 + k + p) > 
  4]

g[p_, k_] := 


1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) ArcCoth[(
       8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p));

nferm[k_]:=1/(Exp[b Sqrt[k^2 + 1]]+1);

Integrate[(k^2 nferm[k])/(2 p k) g[p,k],{k,0,Infinity}]

When I tried to run this, 2 or three minutes of waiting and they just threw back the expression without evaluating it. Can anyone help me with this?

Answer 1

With version 9.0.1,

f[x_] := (p^2 + k^2 - 2 p k x)/(x - (p^2 + k^2 + 1 - ((p^2 - k^2)^2)/4)/(2 p k)); 
ans9 = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
   ArcCoth[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)), 
   k^4 + p^4 < 4 + 4 p^2 + 2 k^2 (2 + p^2) && (k - p)^2 (-2 + k + p) (2 + k + p) > 4] *)

Some insight can be gained by plotting the solution ad its region of validity as specified by ConditionalExpression,

RegionPlot[Evaluate[ans9[[2]]], {k, -5, 5}, {p, -5, 5}, FrameLabel -> {k, p}]

Plot3D[Evaluate[ans9[[1]]], {k, -5, 5}, {p, -5, 5}, PlotPoints -> 100,
  PlotRange -> All, Mesh -> None, AxesLabel -> {k, p, z}]

Thus, the solution is real and continuous over a wider range than that given by the ConditionalExpression. Nonetheless, the actual range is bounded by singularities, so it is not surprising that the second Integrate in the question does not yield an answer. Of course, even without the singularities, Integrate might fail, if no known symbolic solution exists.

Because, I presume, k and p are meant to be Reals, it is reasonable to inform Integrate of this.

ans9r = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True, 
  Assumptions -> k ∈ Reals && p ∈ Reals]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
   ArcTanh[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)),
   (-4 + (k - p)^2) (k + p)^2 <= 4 && (k - p)^2 (-2 + k + p) (2 + k + p) <= 4] *)

which produces a different but related solution (perhaps, a different branch).

Between them, the two solutions cover all of p - k space. Nonetheless, it is unclear whether patching the two together and then integrating would be successful, due to branchpoints at z = 1 and z = -1, where z is the argument of ArcTanh or ArcCoth in the expressions above.

Answer to question in a comment above: Although version 10.1 produces symbolic solutions that appear different from the version 9.0.1 solutions.

(* ConditionalExpression[1/4 (-(-4 + (k^2 - p^2)^2) Log[-k p] + (k^2 - p^2)^2 (Log[k p]
   - Log[4 - (-4 + (k - p)^2) (k + p)^2] + Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])
   - 4 (4 k p + Log[k p] - Log[4 - (-4 + (k - p)^2) (k + p)^2] + 
   Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])),
   (-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))/(k p) ∈ Reals] *)

and

(* ConditionalExpression[1/4 (-16 k p - (-4 + (k^2 - p^2)^2) 
   Log[4 - (-4 + (k - p)^2) (k + p)^2] + (-4 + (k^2 - p^2)^2) 
   Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)]), 
   k^4 + p^4 <= 4 + 4 p^2 + 2 k^2 (2 + p^2) && (-4 + (k - p)^2) (k + p)^2 <= 4] *)

their 3D plots are identical respectively to the fourth and second plots above, although their asserted ranges of validity differ. Even though a straightforward application of FullSimplify does not succeed in showing the respective version 9.0.1 and version 10.1 solutions to be identical, it appears that they are from identities in Wolfram MathWorld.