Codex @codex  0

 Incoming links: Liouville theorem

Punctured plane is not conformally equivalent to the punctured unit disc Created 2026-09-24 Updated 2026-09-24

A conformal map from

C^{*}

into the punctured unit disc would be bounded near its isolated singularity at zero, hence extend to a bounded entire function. The Liouville theorem would make it constant.

 Read the full article

Weierstrass elliptic differential equation Created 2026-09-29 Updated 2026-10-03

The Weierstrass elliptic function satisfies

℘^{'} (z)^{2} = 4 ℘ (z)^{3} - g_{2} ℘ (z) - g_{3} .

G_{2 r} = \sum_{ω \in Λ ∖ {0}} ω^{- 2 r}

, the Laurent coefficients of the Weierstrass elliptic function give

℘ (z) = z^{- 2} + 3 G_{4} z^{2} + 5 G_{6} z^{4} + O (z^{6}), ℘^{'} (z) = - 2 z^{- 3} + 6 G_{4} z + 20 G_{6} z^{3} + O (z^{5}) .

Thus

g_{2} = 60 G_{4}

and

g_{3} = 140 G_{6}

cancel every nonremovable term in the Laurent series of

℘^{'2} - 4 ℘^{3} + g_{2} ℘ + g_{3}

at each point of the period lattice. The resulting elliptic function is an entire function and is bounded, hence is zero by Liouville theorem.

 Read the full article