this is fredholm type of integral equation that im currently triying to solve


the question is that how to derive $\phi(t)$

i tried convolution theorem of fourier transform to solve this equation

and what i got is $ \phi(t)=\sqrt{8\over7\pi}e^{-t^2\over7}$

but it seems suspicious.. is it correct answer?

  • $\begingroup$ Your answer is correct. $\endgroup$ Dec 16, 2015 at 14:10

1 Answer 1


There is an easy solution through trial and error. By completing the square and exploing $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ we have that for any $A\in (-\infty,1)$ $$ \int_{\mathbb{R}}\exp\left(-(t-x)^2+At^2\right)\,dt = \sqrt{\frac{\pi}{1-A}}\exp\left(\frac{Ax}{1-A}\right) $$ holds, hence by solving $\frac{A}{1-A}=\frac{1}{8}$ we get $A=-\frac{1}{7}$, so $$ \phi(t) = \sqrt{\frac{8}{7\pi}}\,e^{-t^2/7} $$ is a solution for sure.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.