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Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$?


$F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and for $a=\dfrac\pi2$ there are $F(x)=\sin x$ and $F(x)=\cos x$. But the three are connected by Euler's formula $e^{ix}=\cos x$ $+i\sin x$. Indeed, on a more general note, letting $F(x)=e^{\lambda x}$, we have $\lambda=\dfrac{W(-a)}{-a}$ where W is the Lambert W function. My question would be if these are the only ones, due to the special properties of the number e and the exponential function, or if there aren't by any chance more, which do not belong in the same family or category as these, i.e., which are not exponential or trigonometric in nature ? Thank you.

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    $\begingroup$ That seems like an unrealistic expectation. Delay differential equations have an entire interval of definition as input. So you could easily ensure that's not an exponential or trig function. $\endgroup$ Commented May 29, 2014 at 0:01
  • $\begingroup$ For example, if $a<0$ is real, look at the solutions with $F(x)=0$ on $[0,-a)$. That's not trig or exponential, unless its the zero function. $\endgroup$ Commented May 29, 2014 at 1:07
  • $\begingroup$ Possibly the question becomes more interesting (or tractable) if $F$ is assumed analytic? If $F$ is merely $C^\infty$, then taking for example $a = 1$ and any $C^\infty$ function $F$ such that $F^{(n)}(0) = 0 = F^{(n)}(1)$ for all $n \geq 0$, the extension of $F$ to any interval $[n-1, n]$, for $n \in \mathbb{Z}$, can be defined recursively by appealing to the functional equation. $\endgroup$
    – user43208
    Commented May 29, 2014 at 1:45
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    $\begingroup$ Questions on this kind of equation are a recurrent theme on this site. It is an example of a linear retarded differential equation (up to a change of sign). There is a survey "A shifted view of fundamental physics" by Atiyah and Moore, available on arXiv 1009 2176. $\endgroup$
    – couperin
    Commented May 29, 2014 at 3:22
  • $\begingroup$ From a more purely mathematical perspective, I was also able to find this. $\endgroup$
    – Lucian
    Commented May 29, 2014 at 4:14

1 Answer 1

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Look at these MO entries: https://mathoverflow.net/questions/114875/on-equation-fz1-fz-fz/114878#114878 , and https://mathoverflow.net/questions/156312/solve-fx-int-x-1x1-ft-mathrmdt/156315#156315 . They contain the answer to your question.

EDIT. To put it shortly, the answer is: "yes" and "no". In exactly the same sense as the answer to a simpler question: "Is every periodic function an exponential/trigonometric sum"? "Yes" for a physicist, and "no" for a mathematician. But every periodic function is a limit of exp/trig sums.

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  • $\begingroup$ So you're saying that exponential and trigonometric functions are indeed the only type of solutions to this problem, or am I misunderstanding something ? $\endgroup$
    – Lucian
    Commented May 29, 2014 at 8:05
  • $\begingroup$ It depends on the exact definition of "trig and exp functions". You can take limits of their linear combinations. With a proper definition of limits all solutions are such limits. The references are given in MO entries that I cited. $\endgroup$ Commented May 29, 2014 at 8:17
  • $\begingroup$ I guess it's a good thing that I nagged my father into teaching me the Cyrillic alphabet when I was a kid. :-) I found the Gelfond reference on page $379-380$. $\endgroup$
    – Lucian
    Commented May 29, 2014 at 12:00
  • $\begingroup$ Also in RAND, on page $45$. $\endgroup$
    – Lucian
    Commented May 29, 2014 at 12:11
  • $\begingroup$ Gelfond is available in English MR0342890. But in Russian it is available FREE. As most Russian math books are. A good justification to learn Cyrillic alphabet:-) $\endgroup$ Commented May 29, 2014 at 14:08

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