Here's one that's more along the lines of your expertise.Originally Posted by CaptainBlack
What functions of t, if any, f(t) have their Laplace transform F(s) equal to
One thing that always fasincated me was how did Euler ever find the Gamma function? There is no way to do it with diffrencial equations or any other methods. He just was creative and was able to think of such a function.Originally Posted by CaptainBlack
Originally Posted by bobbyk
What type of equation is this?
Maybe, you can manipulate it into a diffrencial equation somehow by taking the derivative of both sides. But the problem is that is a definite integral over there. I do not think this is any known type of equation.
For the Fourier transform, there are many though.
I was reading (fooling around actually ) this book about the heat equation, and there was a formula about its Fourier transform. After some calculations, I ended up with
and just when I was getting happy with this, there was a formula concerning the Hermite functions , namely
which means all Hermite functions have this property also. As if an infinite number of solutions were not enough, linearity grants us that every linear combination of these functions has the same property; And continuity grants that, every function in the closure (in say) of these combinations, is again a fixed point...
Thanks for your interest.
Your Fourier self-transforms are beautiful! And, of course, are on the internet. But I've searched for the Laplace self-transforms and haven't found
any, although I'm sure there must be some (other than f(t)=0)