Tagged Questions

55 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
197 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
674 views

Solving an integral equations using fourier transform

I have to solve the equation $\int_0^{\infty} f(x) \cos{(\alpha x)}\, dx=\frac{\sin{\alpha }}{\alpha}$ Using fourier transform. I know this is half of the usual fourier cosine transform, and so ...
319 views

An inverse definite integral problem

I am seeking a function $f(x)$ that satisfies this condition: $\int_{0}^{\infty }f(x)x^ndx=\sqrt{n!}$ where n is an integer. I guess that $f$ will contain $e^{-\alpha x^2}$ as one of its factors, ...
219 views

Gelfand-Levitan-Marchenko equation

how can one solve the integral $$f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$ (1) so $$q(x)= 2\frac{d}{dx}K(x,x)$$ (2) $$-y''(x)+q(x)y(x)=0$$ (3) $$y(0)=0=y(\infty)$$ $q(x)$ here is ...
Can we solve for $a$ in $b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds$
For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both ...