# Weierstrass p-Function Asympototics?

• "pi"mp
In summary, P(u) is a function of a complex variable z that vanishes at u=w1 and is indeterminate at u=0.
"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?

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?

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

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.

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.

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?

## 1. What is the Weierstrass p-Function?

The Weierstrass p-Function is a mathematical function that was discovered by Karl Weierstrass in the 19th century. It is defined as the elliptic analogue of the exponential function and is denoted by Π(z).

## 2. What are the applications of the Weierstrass p-Function?

The Weierstrass p-Function has various applications in mathematics, physics, and engineering. It is used in the study of elliptic curves, number theory, and complex analysis. It is also utilized in the field of signal processing and digital communications.

## 3. What are the asymptotics of the Weierstrass p-Function?

The asymptotics of the Weierstrass p-Function refer to the behavior of the function as the input value approaches certain limits. In particular, the asymptotics of the function can be studied as the imaginary part of the input value approaches infinity or as the real part of the input value approaches certain critical points.

## 4. How can the asymptotics of the Weierstrass p-Function be calculated?

The asymptotics of the Weierstrass p-Function can be calculated using various mathematical techniques such as contour integration, series expansions, and special function identities. These methods allow for the determination of the behavior of the function in different regions of the complex plane.

## 5. What are the implications of the Weierstrass p-Function asymptotics?

The asymptotics of the Weierstrass p-Function have significant implications in the study of complex functions and their behavior. They can provide insights into the convergence properties of series expansions and can also aid in the understanding of the behavior of other mathematical functions that are related to the Weierstrass p-Function.

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