Weierstrass p-function: construction and algebraic identities

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Problem 4.49 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Connection between $\mu(r)$, $\nu(r)$, $M(r)$ and $N(r)$"

For the Weierstrass σ(z)\sigma (z) function, in the case when the period parallelogram is a square with sides of length 11,

N(r)∼πr2,log⁡M(r)∼π2r2, N(r) \sim \pi r^{2}, \qquad \log M(r) \sim \frac{\pi }{2} r^{2},

as rr tends to ∞\infty.

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Remark.
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See Hille, Vol. II pp. 132-133 and Hurwitz-Courant, pp. 174-175.

Problem 29.16 from Шабунин М.И., Половинкин Е.С., Карлов М.И. "Сборник задач по теории функций комплексного переменного" [ШПК15]More problems in Section 29 "The Symmetry Principle"

Let the function f(z)f(z) conformally map the rectangle {z:∣Re(z)∣<h,∣Im(z)∣<h}\left\{ z:\left|\mathrm{Re}\left(z\right)\right| < h,\left|\mathrm{Im}\left(z\right)\right| < h\right\} onto some disk (or half-plane). Prove that the regular continuation of the function f(z)f(z) is a function meromorphic in the entire plane and periodic with periods 4h4 h and 4ih4 i h.

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References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.
[ШПК15] Шабунин М.И., Половинкин Е.С., Карлов М.И. Сборник задач по теории функций комплексного переменного. 4-е изд., БИНОМ. Лаборатория знаний, 2015.