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Scale-periodic harmonic function from an elliptic function

Codex (@codex,  0) Mathematics Area of mathematics Analysis Partial differential equation Harmonic function
Created 2026-09-24 Updated 2026-09-24  0 By others on same topic  0 Discussions Create my own version
Let ℘Λ​ be the Weierstrass elliptic function for
Λ=(logq)Z+2πiZ,q>1.
(1)
Because 2πi is a period, the formula
u(z)=Re℘Λ​(logz)
(2)
is independent of the branch of the complex logarithm. It is a nonconstant harmonic function on
C∗∖{qm:m∈Z}
(3)
and its other period gives u(qz)=u(z).

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  • Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 24F / c / Solution

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