= Solution
Part (b) gives paths with the same endpoint $z$ but integral values
$$
H(z),\qquad H(z)+4K,\qquad H(z)+2iL.
$$
Applying the local inverse and then continuing it shows that
$$
\mathcal H(w+4K)=\mathcal H(w),
\qquad
\mathcal H(w+2iL)=\mathcal H(w).
$$
Since $K,L>0$, the first <period of a function>[period] is real and the second is purely imaginary, so they are <linear independence>[linearly independent] over $\mathbb R$. Hence $\mathcal H$ is an <elliptic function> with two periods
$$
\boxed{4K\quad\hbox{and}\quad2iL}.
$$
Indeed, it is the <Jacobi elliptic sine>, and these generators give its <period lattice of the Jacobi elliptic sine>[period lattice].
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