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$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...

The Mellin transform of a function $f(x)$ can be written as $$ \mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx $$ Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...