Weierstrass's Elliptic Functions - Addition Theorems

Addition Theorems

The Weierstrass elliptic functions have several properties that may be proved:

\det\begin{bmatrix} \wp(z) & \wp'(z) & 1\\ \wp(y) & \wp'(y) & 1\\ \wp(z+y) & -\wp'(z+y) & 1 \end{bmatrix}=0

(a symmetrical version would be

\det\begin{bmatrix} \wp(u) & \wp'(u) & 1\\ \wp(v) & \wp'(v) & 1\\ \wp(w) & \wp'(w) & 1 \end{bmatrix}=0

where u + v + w = 0).

Also

\wp(z+y)=\frac{1}{4} \left\{ \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \right\}^2 -\wp(z)-\wp(y).

and the duplication formula

\wp(2z)= \frac{1}{4}\left\{ \frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z),

unless 2z is a period.

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