All Questions

Consider $f(x)$, a rapidly decreasing function, such that $\int_0^{\infty} f(x)=0$ and for $x$ near zero: $f(x)=O(x^a)$ (wit $a>0$). Can we interchange the sum and integral and write as below: $$\...

Can we interchange the order of integration of following double integral ? $$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$ Where $F(x,y)= \...

Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$ Can we demonstrate that following integral is ...