I'm trying to evaluate the following infinite integral: $\int_{0}^{\infty} x e^{-\alpha x^2}Y_0(\beta x) \,dx$ If we apply the general result in Gradshteyn & Ryzhik (eq. 6.631.2 on page 706), with $\mu=1$ and $\nu=0$ then the result should be $\int_{0}^{\infty} x e^{-\alpha x^2}Y_0(\beta x) ...
In my course of work I came across the following elliptic integral, and found its solution in Gradshteyn and Ryzhik: $$\int_{b}^{u} \frac{\mathrm{d} x}{\sqrt{(a-x)(x-b)(x-c)(x-d)}} = \frac{2}{\sqrt{(a-c)(b-d)}}\,F(\lambda, r),$$ where $$\lambda = \arcsin \left(\sqrt{\frac{(a-c)(u-b)}{(a-b)(u-c)}...
Preamble I've stumbled upon (a particular variant) of Fibonacci polynomials in my work. The polynomials I'm encountering are of the form $$ F_k(x) = \begin{cases} \qquad\qquad 1, & k=1 \\ \qquad\qquad 1, & k=2 \\ F_{k-1}(x) + x\!\; F_{k-2}(x), & k > 2 \end{cases} $$ giving rise to a sequence of ...
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