Category Archives: Integrals

N.L.R. Rojas and E.A. Galapon, “Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions,” arXiv:2409.10948

By | September 19, 2024
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We perform an asymptotic evaluation of the Hankel transform, $\int_0^{\infty}J_{\nu}(\lambda x) f(x)\mathrm{d}x$, for arbitrarily large $\lambda$ of an entire exponential type function, $f(x)$, of type $\tau$ by shifting the contour of integration in the complex plane. Under the situation that $J_{\nu}(\lambda x)f(x)$ has an odd parity with respect to $x$ and the condition that the… Read More »

A family of integral representations for the reciprocal of Pi

By | May 11, 2022
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Theorem 1. The reciprocal of $\pi$ assumes the following family of integral representations \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+1}}{(2m)!\, 2^{4m+2n}} \int_0^{\infty} \frac{x^{2m} J_{2m+1}(\lambda x)}{\Gamma\left(m+ n+\frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} \begin{equation} \frac{1}{\pi}= \frac{\left[\prod_{k=1}^{m+n}(2k-1)\right]^2 \lambda^{2m+2}}{(2m+1)!\, 2^{4m+2n+1}} \int_0^{\infty} \frac{x^{2m+1} J_{2m+2}(\lambda x)}{\Gamma\left(m+n+ \frac{1}{2} + x\right) \Gamma\left(m+n+\frac{1}{2}-x\right)}\,{\rm d}x \end{equation} for $m=0, 1, 2, \dots$, $n=1, 2, 3,\dots$ and $\lambda\geq\pi$, where $J_n(x)$… Read More »

The analytic principal value by way of the Cauchy-Plemelj-Fox theorem

By | May 5, 2016
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In the publication found here or here, I introduced the concept of analytic principal value (APV) to allow meaningful assignment of values to the class of divergent integrals given by \begin{equation}\label{divergent} \int_a^b\frac{f(x)}{(x-x_0)^{n+1}}\mathrm{d}x, \; \; n=0, 1, 2, \dots . \end{equation}with $f(x_0)\neq 0$. The basic assumption in the definition of the APV is that the function… Read More »

I think this one is beautiful

By | September 14, 2015
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Beauty is one of those abstractions that is difficult to define or quantify. But we know it when we see one. I arrived at the following integral trying to make sense of some divergent integrals: \begin{equation} \frac{1}{2}\int_{-\infty}^{\infty} \left(\cos\left(\frac{t^{-1}-t}{2}\right) – t \sin\left(\frac{t^{-1}-t}{2}\right)\right) (1+t^2)^{-1} \, \mbox{d}t = \frac{\pi}{\mbox{e}} .\nonumber \end{equation} There is manifest symmetry in the integrand… Read More »