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Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...

I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...

Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where \begin{eqnarray}\label{eqn:constraint1} f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...

Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia Frullani integral and other uses and contexts (see [2]). I wrote two imaginative examples of what can be ...