I was toying with equations of the type f(x+α)=f′(x) where f is a real function....
kuhse4461a
Answered
2022-01-15
I was toying with equations of the type where f is a real function. For example if then the solutions include the function . Are there more solutions? On the other hand, if I want to solve the equation for any , I can assume a solution of the form , and find as a complex number that enables me to solve the equation... I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator adds another one? Is there some litterature about this kind of equations?
Answer & Explanation
Melissa Moore
Expert
2022-01-16Added 32 answers
This sort of thing is known as a delay differential equation, and there are lots and lots of books on this subject. They're called ''delay'' because the constant is usually chosen to be negative, so that f′(x) depends on the values of f at earlier times. The equation you have given has an infinite dimensional space of solutions. In particular, if is any smooth function satisfying for all , then there exists a unique extension of g to a smooth solution f defined on the entire real line. For , this solution is defined by where while for it involves iterated integrals of g.
Esther Phillips
Expert
2022-01-17Added 34 answers
Integers without large
alenahelenash
Expert
2022-01-24Added 366 answers
Note that is solution of , and this equation is satisfied by infinitely many complex , namely where are the branches of Lambert