12 Illustration of Formulas Derived From Convolutions

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Note the Fourier series representation of $U'(x)$ converges to $\delta(x-1)$ in the right-half plane and the Mellin transform $\mathcal{M}_x[\delta(x-1)](s)=1$. I’m currently investigating convergence of formulas derived from the Mellin convolutions such as those defined in (1) to (3) below where the Fourier series representation of $U'(x)$ is substituted for $\delta(x-1)$ in the convolutions.

(1) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_1}\,g(x)=\int_0^\infty\delta(x-1)\,g\left(\frac{y}{x}\right)\,\frac{dx}{x}$

(2) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_2}\,g(x)=\int_0^\infty\delta(x-1)\,g(y\,x)\,\,dx$

(3) $\quad g(y)=\delta(x-1)\,*_\mathcal{M_3}\,g(x)=\int_0^\infty\delta(x-1)\,g(y+1-x)\,\,dx$

It’s also possible to derive formulas for derivatives of functions using the Fourier series representations of the derivatives of $\delta(x-1)$ via relationships such as the following.

(4) $\quad g^{(n)}(y)=(-y)^{-n}\left(\delta^{(n)}(x-1)\,*_{\mathcal{M}_2}\,g(x)\right)=(-y)^{-n}\int_0^\infty\delta^{(n)}(x-1)\,g(y\,x)\,dx$

(5) $\quad g^{(n)}(y)=\delta^{(n)}(x-1)\,*_{\mathcal{M}_3}\,g(x)=\int_0^\infty\delta^{(n)}(x-1)\,g(y+1-x)\,dx$

An initial formula derived via a convolution such as those described above may be used to derive additional formulas for related functions via differentiation, integration, Mellin normal/inverse transforms, Laplace normal/inverse transforms, and Hankel transforms.

I’m currently investigating convergence of a number of new formulas derived from the convolutions described above for a variety of functions such as $x^j$, $\frac{1}{x+a}$, $log[x]$, $e^{-x}$, $\sin[x]$, $\cos[x]$, $Ei[x]$, $E_n[x]$, $\Gamma[a,\,x]$, $\frac{1}{\Gamma[x]}$, and various $Bessel$ functions. I plan to illustrate several of these formulas on children pages below when I find time, but right now my priority is derivation of new formulas from convolutions such as those defined above and investigation of the convergence of these formulas.