Incorrect limit in complex quantity

Question

I have the following quantity

$$ f = \frac{\exp(\sqrt{-k^2-i\epsilon}d)}{\sqrt{-k^2-i\epsilon}} $$

and form the limit as $\epsilon\to0$. Mathematica does not produce the correct result.

Code:

f = -(E^(Sqrt[-k^2 - I ϵ] d)/Sqrt[-k^2 - I ϵ]);

g1 = Limit[f, ϵ -> 0, Direction -> "FromAbove", 
   Assumptions -> Element[{k}, Reals] && ϵ >= 0];
g2 = Limit[f, ϵ -> 0, Direction -> "FromAbove", 
   Assumptions -> Element[{k}, Reals] && ϵ >= 0 && k > 0];

The second one gives the correct result for positive $k$, the first one yields the complex conjugate. Of course this is related to various branch cut things, but I am very surprised that the assumption $k>0$ would yield any difference, since only $k^2$ appears in the equation.

By "correct result" I mean the limit as $\epsilon\to0^+$. I can take e.g. $\epsilon=10^{-10}$ and get a result that agrees with the $g_2$ up to errors of $\mathcal O(10^{-10})$, but not with $g_1$.

Answer 1

This is definitely a bug, apparently Limit is playing fast and loose with exponentials of logs. I will file a bug. But the correct answer is to take the g1 and replace Abs[k] with -Abs[k].

FullSimplify[(g1 /. Abs[k] -> -Abs[k]) == g2, k > 0]
(*True*)

And if you defined g3 for k<0, you also find:

FullSimplify[(g1 /. Abs[k] -> -Abs[k]) == g3, k < 0]
(*True*)

BTW, you don't need the assumption k\[Element]Reals in g2 and g3. If you put an assumption with an inequality, both sides are assumed to be real.

Answer 2

The solution with Conjugate ... takes into acount, that k can be zero.

Let me give you an overview, that all solutions are true.

f = -(E^(Sqrt[-k^2 - I \[Epsilon]] d)/Sqrt[-k^2 - I \[Epsilon]]);

{{" ", " ", "k>0", "k<0"}, {"Direction\[Rule]1,k\[Element]Reals", 
g1 = Limit[f, \[Epsilon] -> 0, Direction -> 1, 
 Assumptions -> Element[{k}, Reals]], Simplify[g1, k > 0], 
Simplify[g1, k < 0]}, {"Direction\[Rule]-1,k\[Element]Reals", 
g2 = Limit[f, \[Epsilon] -> 0, Direction -> -1, 
 Assumptions -> Element[{k}, Reals]], Simplify[g2, k > 0], 
Simplify[g2, k < 0]}, {"Direction\[Rule]1,k\[Element]Reals,k!=0", 
g3 = Limit[f, \[Epsilon] -> 0, Direction -> 1, 
 Assumptions -> Element[{k}, Reals] && k != 0], 
Simplify[g3, k > 0], 
Simplify[g3, k < 0]}, {"Direction\[Rule]-1,k\[Element]Reals,k!=0", 
g4 = Limit[f, \[Epsilon] -> 0, Direction -> -1, 
 Assumptions -> Element[{k}, Reals] && k != 0], 
Simplify[g4, k > 0], Simplify[g4, k < 0]}, {"Direction\[Rule]1", " ",
 g5 = Limit[f, \[Epsilon] -> 0, Direction -> 1, 
 Assumptions -> k > 0], 
 g6 = Limit[f, \[Epsilon] -> 0, Direction -> 1, 
 Assumptions -> k < 0]}, {"Direction\[Rule]-1", " ", 
 g7 = Limit[f, \[Epsilon] -> 0, Direction -> -1, 
 Assumptions -> k > 0], 
 g8 = Limit[f, \[Epsilon] -> 0, Direction -> -1, 
 Assumptions -> k < 0]}} // Grid[#, Frame -> All] &

Answer 3

You deal with a complex-valued function so the Direction->Complexes option does the job:

Limit[Exp[Sqrt[-k^2 - I*\[Epsilon]]*d]/Sqrt[-k^2 - I*\[Epsilon]], 
\[Epsilon] -> 0, Direction -> Complexes,GenerateConditions->True]
(* ConditionalExpression[E^(d Sqrt[-k^2])/Sqrt[-k^2], k != 0 && Im[k^2] != 0]*)

Addition. The same result is produced by

Limit[Exp[Sqrt[-k^2 - I*\[Epsilon]]*d]/Sqrt[-k^2 - I*\[Epsilon]], \[Epsilon] -> 0,GenerateConditions->True]