Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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|improve this question
add comment

1 Answer

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

Your Answer

 
discard

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