Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below:
$$f'(x)=f(x+c),$$
where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.


I found two families of functions (i.e. exponential functions and sine/cosine functions) that satisfy this differential equation. For example, $f(x)=c\_1 \sin(x+c\_2)$ and $f(x)=c\_1 e^x$.


Can you come up with any more function families or prove no more functions exists?



One question about a specific first-order differential equation
Following Euler, let us look at solutions of the form $f(z)=e^{sz}$, where $s$ is a complex number. We obtain a transcendental equation
$$s=e^{cs},$$
which for every complex $c\neq 0$ has infinitely many complex solutions $s\_n$. Now any linear combination
$$\sum\_{n}a\_ne^{s\_nz}$$
with complex $a\_n$, and any limit of such linear combinations will satisfy your equation. This gives all solutions.


The method actually applies to a wide class of linear equations of "convolution type". See
Malgrange's approximation theorem (Theorem 16.4.1 in Hormander, Analysis of linear partial differential operators) and also these entries on MO: [Solve $f(x)=\int\_{x-1}^{x+1} f(t) \mathrm{d}t$](https://mathoverflow.net/questions/156312/),
[On equation $f(z+1)-f(z)=f'(z)$](https://mathoverflow.net/questions/114875/)


