Ballooning transformation

To construct a periodic function about $\theta$, we introduces a function $\overline{z} (\theta)$ which is defined over $- \infty < \theta < \infty$ and vanishes sufficiently fast as $\vert \theta \vert \rightarrow \infty$ so that the following infinite summation converge:

$$\sum_{l = - \infty}^{\infty} \overline{z} (\theta + 2 \pi l) .$$ (524)

If we use the above sum to define a function

$$z (\theta) = \sum_{l = - \infty}^{\infty} \overline{z} (\theta + 2 \pi l),$$ (525)

then it is obvious that

$$z (\theta + 2 \pi) = z (\theta),$$ (526)

i.e., $z (\theta)$ is a periodic function about $\theta$ with period of $2 \pi$.

If we use the right-hand-side of Eq. (525) to represent $z (\theta)$, then we do not need to worry about the periodic property of $z (\theta)$ (the periodic property is guaranteed by the representation)