Justify if $\frac{\sin(x)\sin(y)}{x^2 + y^2}$ is integrable over $ (-\frac{\pi}{2}, \frac{\pi}{2})\times (-\frac{\pi}{2}, \frac{\pi}{2})$.

Question

Justify if $\frac{\sin(x)\sin(y)}{x^2 + y^2}$ is integrable over $ (-\frac{\pi}{2}, \frac{\pi}{2})\times (-\frac{\pi}{2}, \frac{\pi}{2})$.

I aim to use Tonelli's theorem, but am not sure where to start. An attempt to use a x=ty (with y fixed) substitution did not work. Any help would be much appreciated, thank you.

Absolute value is less than 1/(y^2). Clearly that will give a finite integral when integrated over the space — latbbltes, Apr 09 '22 at 21:14
@SacAndSac Ah yeah you're right im being dumb because 1/y^2 is obviously gonna be infinite when integrated — latbbltes, Apr 09 '22 at 21:21

score 3 · Answer 1 · answered Apr 09 '22 at 21:29

3

Note that $|\sin x| \leq |x| \leq \sqrt{x^2 + y^2}$ and similarly $|\sin y| \leq |y| \leq \sqrt{x^2 + y^2}$, therefore $$ \left|\frac{\sin x \sin y}{x^2 + y^2}\right| \leq \frac{x^2+y^2}{x^2+y^2} = 1 $$ for $(x,y) \neq (0,0)$. Thus the function is bounded in absolute value by $1$ and has at most one discontinuity at $(0,0)$, which is irrelevant for integrability. Therefore this function is integrable.

answered Apr 09 '22 at 21:29

nullUser

27,877

thank you very much – user2080 Apr 09 '22 at 21:45

Justify if $\frac{\sin(x)\sin(y)}{x^2 + y^2}$ is integrable over $ (-\frac{\pi}{2}, \frac{\pi}{2})\times (-\frac{\pi}{2}, \frac{\pi}{2})$.

1 Answers1