- #1
Whovian
- 652
- 3
Firstly, if this is an inappropriate forum for this thread, feel free to move it. This is a calculus-y equation related to differential equations, but I don't believe it's strictly a differential equation.
The question I'm asking is which functions ##f:\left[0,\infty\right)\rightarrow\mathbb{R}## and real constants ##\lambda## have the property that ##\int_0^\infty\left(f\left(t\right)\cdot e^{-s\cdot t}\right)\cdot\mathrm{d}t=f\left(s\right)## for all ##s## in some open interval.
The question was left somewhat open-ended in this old thread, but since it was from 6 years ago, I felt reviving it would be somewhat unnecessary.
Induction on ##n## gives us the apparently trivial condition that ##\int_0^\infty\left(\left(-t\right)^n\cdot e^{-s\cdot t}\cdot f\left(t\right)\right)\cdot\mathrm{d}t=\lambda\cdot f^{\left(n\right)}\left(s\right)##; the left hand side seems to be screaming Caputo fractional derivative, so perhaps this is of some use. That's basically all I've got.
The question I'm asking is which functions ##f:\left[0,\infty\right)\rightarrow\mathbb{R}## and real constants ##\lambda## have the property that ##\int_0^\infty\left(f\left(t\right)\cdot e^{-s\cdot t}\right)\cdot\mathrm{d}t=f\left(s\right)## for all ##s## in some open interval.
The question was left somewhat open-ended in this old thread, but since it was from 6 years ago, I felt reviving it would be somewhat unnecessary.
Induction on ##n## gives us the apparently trivial condition that ##\int_0^\infty\left(\left(-t\right)^n\cdot e^{-s\cdot t}\cdot f\left(t\right)\right)\cdot\mathrm{d}t=\lambda\cdot f^{\left(n\right)}\left(s\right)##; the left hand side seems to be screaming Caputo fractional derivative, so perhaps this is of some use. That's basically all I've got.