For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$?

Question

The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values of a and b make $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$. The trouble now is I'm not clear on how to check this inequality over all of $\mathbb{R}$. Actually, guessing and checking hasn't found me a single pair of values for which the integral isn't finite, which I'm sure would be far too easy and anyway I wouldn't know how to prove it.

I just need a hint on how to rigorously check the finiteness of this integral, or if this is the wrong approach to the problem, what the right approach might be.

Answer 1

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left( #1 \right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r < \infty:\ {\large ?}}$

With $\ds{\quad r \equiv \expo{t}\quad\imp\quad t = \ln\pars{r}}$\begin{align} &\color{#c00000}{\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r} =-\int_{\infty}^{2}r^{- a - n - 1}\ln^{b}\pars{r}\,\dd r =\int_{\ln\pars{2}}^{\infty}\expo{-\pars{a + n + 1}t}t^{b}\pars{\expo{t}\,\dd t} \\[3mm]&=\int_{\ln\pars{2}}^{\infty}t^{b}\expo{-\pars{a + n}t}\,\dd t \quad\mbox{which converges when}\quad\pars{a + n} > 0\quad\mbox{and}\quad b > -1 \end{align}

\begin{align} &\color{#00f}{\large\int_{0}^{1/2}r^{a + n - 1}\verts{\ln\pars{r}}^{b}\,\dd r} ={1 \over \pars{a + n}^{b + 1}}\int_{\pars{a + n}\ln\pars{2}}^{\infty}t^{b} \expo{-t}\,\dd t \\[3mm]&=\color{#00f}{\large% {\Gamma\pars{b + 1,\pars{a + n}\ln\pars{2}} \over \pars{a + n}^{b + 1}}}\,,\qquad \pars{a + n} > 0\,,\quad b > - 1 \end{align} where $\ds{\Gamma\pars{s,x}}$ is the Incomplete Gamma Function.