Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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 $z$.

share|cite|improve this answer
Indeed, silly me. Thank you! – user519 Sep 17 '12 at 22:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.