How can I evaluate the following integral, for $\lambda >0$ and $0 \le k \le 1-\epsilon$, with $\epsilon>0$ arbitrarily small number?
$$I = \int_{\mathbb{R}}\frac{e^{-\lambda |x|}}{|x|^k}dx$$
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2Hint:$$\int_{\mathbb R}\frac{e^{-\lambda|x|}}{|x|^k}\ dx=2\int_0^\infty x^{-k}e^{-\lambda x}\ dx$$Let $u=\lambda x$ and the Gamma function. – Simply Beautiful Art May 16 '17 at 16:49
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Just exploit parity, the substitution $x=\frac{z}{\lambda}$ and the integral definition of the $\Gamma$ function.
It turns out that
$$\int_{\mathbb{R}}\frac{e^{-\lambda|x|}}{|x|^k}\,dx = \color{red}{2\lambda^{k-1}\Gamma(1-k)}$$ for any $\lambda>0$ and $k\in(0,1)$. If $1-k=\varepsilon$ is positive and close to zero, the given integral is $\approx\color{red}{ \frac{2}{\varepsilon \lambda^\varepsilon}}$, since the $\Gamma$ function has a pole with residue $1$ at the origin.
Jack D'Aurizio
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Thank you very much for your answer. I'm not quite familiar with formulas involving the Gamma function. Would you mind writing down some additional details for your claims? – May 16 '17 at 16:55
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@Hideki: everything is pretty well-known: https://en.wikipedia.org/wiki/Gamma_function – Jack D'Aurizio May 16 '17 at 16:56
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Also, is are there closed formulas to compute the numerical value of the integral? – May 16 '17 at 16:57
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@Hideki: any CAS is able to compute $\Gamma(x)$ for some $x\in\mathbb{R}^+$ with an arbitrary degree of accuracy. It is enough to exploit Euler's or Weierstrass' products, for instance. Again, the best reference is https://en.wikipedia.org/wiki/Gamma_function. – Jack D'Aurizio May 16 '17 at 16:59
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@Hideki Note that $\Gamma(n+1)=n!$, and see here for half integers and see here for other rational values. – Simply Beautiful Art May 16 '17 at 17:05
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@Hideki: the $\sup$ when $k$ ranges over ? Such quantity is clearly unbounded when $k\to 1^-$. – Jack D'Aurizio May 16 '17 at 17:28
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@Hideki: the supremum over $(0,1)$ is $+\infty$ as a straighforward consequence of the shown approximation, the supremum over $[0,1-\varepsilon]$ is the maximum achieved at the right endpoint. – Jack D'Aurizio May 16 '17 at 17:36
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That's right. Thanks a lot. Do you also happen to know what input should I give to Mathematica or SageMath or Matlab to have a plot the value of the integral for various values of $0≤k<1$ and $λ>0$? – May 16 '17 at 17:42
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@Hideki: in Mathematica, $$ \text{Plot[Evaluate[Table[2*Gamma[1 - k]/(n/10)^(k - 1), {n, 10}]], {k, 0, 1}]}$$ – Jack D'Aurizio May 16 '17 at 17:52