I am attempting to solve the equation
$$f(x + 1) = f'(x)$$
for distributions $C \rightarrow C: f(x)$
My first guess to exploit the fact that this seems similar to identity
$$\sin\left( \frac{\pi}{2} - x\right) = \cos(x) = \frac{d}{dx} [\sin(x)] $$
I therefore assume my answer takes on the form:
$$Ce^{ax}$$
And now attempt to solve:
$$Ce^{a(x+1)} = Cae^{ax}$$
Yielding
$$e^{a}e^{ax} = ae^{ax}$$
Yielding
$$e^a = a$$
How do I extract a single solution, let alone the infinitely many complex solutions for $a$ that satisfy this equation?
