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Part (b) gives paths with the same endpoint $z$ but integral valuesApplying the local inverse and then continuing it shows thatSince $K,L>0$, the first period is real and the second is purely imaginary, so they are linearly independent over $\mathbb{R}$. Hence $\mathcal{H}$ is an elliptic function with two periodsIndeed, it is the Jacobi elliptic sine, and these generators give its period lattice.