# Unexpected behavior in symbolic integration with GenerateConditions->False

Consider the following two symbolic integrations:

int[a_, z_] :=
HeavisideTheta[-t^2 + r^2 + r0^2 - 2 r*r0* z] LegendreP[a, z]
Table[Integrate[int[a, z], {z, -1, 1},
Assumptions -> {t > 0 && r > 0 && r0 > 0},
GenerateConditions -> True], {a, 0, 3}]


and

int[a_, z_] :=
HeavisideTheta[-t^2 + r^2 + r0^2 - 2 r*r0* z] LegendreP[a, z]
Table[Integrate[int[a, z], {z, -1, 1},
Assumptions -> {t > 0 && r > 0 && r0 > 0},
GenerateConditions -> False], {a, 0, 3}]


In the first I set GenerateConditions->True in the second False.

The second gives 0 for each a, while the first gives a long result which is (very!) different from 0 (and no conditions btw).

Why is it so? I am using version 10 for Windows

Edit This is the output of the first code:

{(HeavisideTheta[((r + r0 - t) (r + r0 + t))/(r r0)] ((r + r0 - t) (r + r0 + t) - ((r - r0)^2 - t^2) HeavisideTheta[((r - r0)^2 - t^2)/(r r0)]))/(2 r r0), -(((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t) HeavisideTheta[((r + r0 - t) (r + r0 + t))/( r r0)] (-1 + HeavisideTheta[((r - r0)^2 - t^2)/(r r0)]))/(8 r^2 r0^2)), -(((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t) (r^2 + r0^2 - t^2) HeavisideTheta[((r + r0 - t) (r + r0 + t))/( r r0)] (-1 + HeavisideTheta[((r - r0)^2 - t^2)/(r r0)]))/(16 r^3 r0^3)), -(((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t) (5 r^4 + 2 r^2 (3 r0^2 - 5 t^2) + 5 (r0^2 - t^2)^2) HeavisideTheta[((r + r0 - t) (r + r0 + t))/( r r0)] (-1 + HeavisideTheta[((r - r0)^2 - t^2)/(r r0)]))/(128 r^4 r0^4))}

• What happens when you apply FullSimplify[] to the output of the first snippet? Commented Feb 27, 2016 at 13:12
• It returns the input Commented Feb 27, 2016 at 13:30
• Nothing in the "long result" changed at all? Commented Feb 27, 2016 at 13:31
• no, nothing changes Commented Feb 27, 2016 at 13:37
• Try putting your assumptions as the second argument of FullSimplify[]. Commented Feb 27, 2016 at 14:20

Both answers given in the question are correct, consistent with the documentation of Integrate. To see this, determine the value of z for which HeavisideTheta changes in value from 1 to 0.

z0 = z /. Solve[-t^2 + r^2 + r0^2 - 2 r*r0*z == 0, z][[1, 1]]
(* (r^2 + r0^2 - t^2)/(2 r r0) *)


which also can be written as 1 + ((r - r0)^2 - t^2)/(2 r r0). Then, labeling as s the general solution provided in the edit of the question,

FullSimplify[s, z0 < -1]
{0, 0, 0, 0}


According to the Options>GenerateConditions section of Integrate, setting GenerateConditions -> False will "Generate a result without conditions that is valid only for some values of parameters", which we see is indeed the case. For completeness,

FullSimplify[s, z0 > 1]
(* {2, 0, 0, 0} *)

FullSimplify[s, -1 < z0 < 1]
(* {((r + r0 - t) (r + r0 + t))/(2 r r0),
((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t))/(8 r^2 r0^2),
((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t) (r^2 + r0^2 - t^2))/(16 r^3 r0^3),
((r - r0 - t) (r + r0 - t) (r - r0 + t) (r + r0 + t) (5 r^4 + 2 r^2 (3 r0^2 - 5 t^2)
+ 5 (r0^2 - t^2)^2))/(128 r^4 r0^4)} *)


An alternative way of looking at this problem is to rewrite int as

int[a_, z_] := HeavisideTheta[z0 - z] LegendreP[a, z]


Thus, for z0 < -1 int is 0 throughout the range of integration, {z, -1, 1}, so Integrate must yield 0 for all integer a. For z0 > -1, on the other hand, the integrals take the form,

Table[Integrate[LegendreP[a, z] , {z, -1, Min[1, z0]}, Assumptions -> z0 > -1], {a, 0, 3}]
(* {1 + Min[1, (r^2 + r0^2 - t^2)/(2 r r0)],
-(1/2) + 1/2 Min[1, (r^2 + r0^2 - t^2)/(2 r r0)]^2,
-(1/2) Min[1, (r^2 + r0^2 - t^2)/(2 r r0)] + 1/2 Min[1, (r^2 + r0^2 - t^2)/(2 r r0)]^3,
1/8 - 3/4 Min[1, (r^2 + r0^2 - t^2)/(2 r r0)]^2 +
5/8 Min[1, (r^2 + r0^2 - t^2)/(2 r r0)]^4} *)


equivalent to s for z > -1.