Weierstrass's Elliptic Functions - Differential Equation

Differential Equation

With this notation, the ℘ function satisfies the following differential equation:

where dependence on and is suppressed.

This relation can be quickly verified by comparing the poles of both sides, for example, the pole at z = 0 of lhs is

^2|_{z=0}\sim \frac{4}{z^6}-\frac{24}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}-80\sum \frac{1}{(m\omega_1+n\omega_2)^6}

while the pole at z = 0 of

^3|_{z=0}\sim \frac{1}{z^6}+\frac{9}{z^2}\sum \frac{1}{(m\omega_1+n\omega_2)^4}+15\sum \frac{1}{(m\omega_1+n\omega_2)^6}.

Comparing these two yields the relation above.

Read more about this topic:  Weierstrass's Elliptic Functions

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