I have a uniform probability distribution in polar coordinates for r
as well as phi
. To get the marginal distribution as a function of x
, I have to intgrate:
r[x_, y_] := Sqrt[x^2 + y^2];
p[x_, y_] := ArcTan[x, y];
G[r_, p_] :=
1/(b - a)*1/(d - c)*Boole[r - a >= 0]*Boole[b - r >= 0]*
Boole[p - c >= 0]*Boole[d - p >= 0]
Integrate[G[r[x, y], p[x, y]]/r[x, y], {y, -Infinity, Infinity},
Assumptions -> {a > 0, b > 0, b > a, c > 0, d > 0, d > c,
Element[x, Reals], Element[y, Reals]}]
However, this integral does not evaluate. If I leave out the condition on the angle p
in the definition of G
, everything works fine. Does the integral simple get to complicated or can this be solved somehow?
Boole
functions as integral boundaries? $\endgroup$1/Sqrt[x^2+y^2]
can be solved. The Boolean functions can be transferred to the boundaries. Thus, the integral should have a closed form on different intervals of x. $\endgroup$