I'm having a problem getting Mathematica in vanishing the following expression:
$\frac{-3 \:\sqrt{6}\: x \sqrt{\frac{y}{3 x-2 \sqrt{3}}}+ \:6 \sqrt{2} \: \sqrt{\frac{y}{3 x-2 \sqrt{3}}}+\: \sqrt{6}\: \sqrt{\left(3 x-2 \sqrt{3}\right) y}}{12 \sqrt{\pi } \:x}$
When I plug it into Mathematica:
FullSimplify[((
6 Sqrt[2] Sqrt[-(y/(2 Sqrt[3] - 3 x))] -
3 Sqrt[6] x Sqrt[-(y/(2 Sqrt[3] - 3 x))] +
Sqrt[6] Sqrt[(-2 Sqrt[3] + 3 x) y])/(12 Sqrt[\[Pi]] x))]
it obviously cannot cancel the terms since I have not specified what the forms x and y take (real, complex, positive etc.).
Now I know the result is zero for real values of $x>\frac{\sqrt{3}}{2}$ and $y\neq0$. I can actually get Mathematica to give me these exact conditions by using Reduce:
Reduce[{(6 Sqrt[2] Sqrt[-(y/(2 Sqrt[3] - 3 x))] -
3 Sqrt[6] x Sqrt[-(y/(2 Sqrt[3] - 3 x))] +
Sqrt[6] Sqrt[(-2 Sqrt[3] + 3 x) y])/(12 Sqrt[\[Pi]] x) == 0,
y > 0, x \[Element] Integers}, {x, y}, Reals]
I have added the condition $x \in \mathbb{Z}$ which then gives me the correct answer:
x \[Element] Integers && x >= 2 && y > 0
Perfect! These are the conditions I need to specify to FullSimplify
for the cancelling to occur. However, when I try
FullSimplify[(
6 Sqrt[2] Sqrt[-(y/(2 Sqrt[3] - 3 x))] -
3 Sqrt[6] x Sqrt[-(y/(2 Sqrt[3] - 3 x))] +
Sqrt[6] Sqrt[(-2 Sqrt[3] + 3 x) y])/(12 Sqrt[\[Pi]] x),
x \[Element] Integers && x >= 2 && y > 0]
it still is not able to give me 0 as an answer, it just spits out the answer again. What exactly is happening here? I'm at my wits' end.