I'm attempting some questions from Zwiebach – A First Course in String Theory, and have got stuck. I've proved that a function $h'(u)$ is periodic. The question then asks me to show that $h(u)=au+f(u)$ where $a$ is a constant and $f(u)$ a periodic function. I can't see how to do this directly from the periodicity of $h'$. Is this possible, or true?
Many thanks!
Best Answer
We may assume without loss of generality that the period of $h'$ is $1$, so that $h'(u + 1) = h'(u)$.
Consider $h(u + 1) - h(u)$. By differentiation, we find that $h(u + 1) - h(u) = a$ for some constant $a$. One now guesses that $h(u) = a u + f(u)$ for some periodic function $f$. So, $$f(u + 1) - f(u) = h(u + 1) - h(u) - a = 0$$ hence $f$ is indeed periodic with period $1$.