# Solution of $z(t+a) = h(a)z(t)$

I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation

$$z(t+a) = h(a)z(t)$$

he makes the following derivation (I'm citing the book):

If we differentiate both sides of [the aforementioned equation] with respect to $a$, we obtain $$z'(t+a) = h'(a)z(t).$$ Setting $a = 0$ gives $$z'(t) = Hz(t)$$ where $$H \equiv h'(0).$$ This implies $$z(t) \propto e^{Ht}.$$ Thus the irreducible solution is an exponential! [...]

How can he differentiate with respect to $a$ then solve the resulting differential equation with respect to $t$? This makes no sense to me.

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The key is to realize that $$\frac{d z(t+a)}{d a}=\frac{d z(t+a)}{dt}$$ since changing $a$ and changing $t$ produce the same change in the argument of $z$.