Weierstrass p-Function Asympototics?

["pi"mp]

Hi,
I need to study the function:
$$\bigg| \wp(u ; g_{2}, g_{3})- \wp( (u+2 \omega_{1}); g_{2}, g_{3}) \bigg|$$

where $u$ is the real part of the argument and I'm using the convention where $\omega_{1}$ is actually half of the overall period on the torus.

Specifically, I'd like asymptotics for both small $\omega_{1}$ and large $\omega_{1}$. I haven't been able to find anything too helpful in the form of addition formulas or anything.

Has anyone seen anything potentially helpful in any of the literature?
Thanks in advance!

 

[mathwonk]

P is a function of a complex variable z. What do you mean that u is the real part of the argument? u = Re(z)? Then the meaning of P(u) is not clear to me. Do you mean you are looking at P(z) only for real values of the argument z?

 

["pi"mp]

That is indeed what I meant. Sorry. I only care about u real from zero to the half-period.

 

[mathwonk]

why isn't the function you wrote down identically zero? I.e. you are adding a period to u and hence subtracting the same value.

 

["pi"mp]

Ugh, you're right...stupid typo I meant to write the function:

$$\bigg| \wp(u) - \wp(2 \omega_{1} - u) \bigg|$$

So it vanishes at u=w1 and is indeterminate at u=0. I did find the following formula in terms of Weierstrass Sigma Functions:

$$\wp(z_{1})-\wp(z_{2}) = \frac{\sigma(z_{1}+z_{2}) \sigma(z_{1}-z_{2})}{\sigma(z_{1})^{2} \sigma(z_{2})^{2}}$$

however it seems like the infinite product which defines the sigma function will be a pain. Maybe there are asymptotics on the sigma function I can look at.

 

["pi"mp]

Wait...now I'm confusing myself. The formula with the absolute value that I wrote just above should also vanish right? Since Weierstrass P is even! However, I have plots in mathematica where the absolute value is clearly not zero. I've also made sure that my half-periods agree with my g2, g3 values. What's going on here?