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Weierstrass p-Function Asympototics?

  1. Jul 16, 2015 #1
    Hi,
    I need to study the function:
    [tex] \bigg| \wp(u ; g_{2}, g_{3})- \wp( (u+2 \omega_{1}); g_{2}, g_{3}) \bigg| [/tex]

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

    Specifically, I'd like asymptotics for both small [itex] \omega_{1} [/itex] and large [itex] \omega_{1} [/itex]. 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!
     
  2. jcsd
  3. Jul 16, 2015 #2

    mathwonk

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    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?
     
  4. Jul 16, 2015 #3
    That is indeed what I meant. Sorry. I only care about u real from zero to the half-period.
     
  5. Jul 16, 2015 #4

    mathwonk

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    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.
     
  6. Jul 16, 2015 #5
    Ugh, you're right...stupid typo I meant to write the function:

    [tex] \bigg| \wp(u) - \wp(2 \omega_{1} - u) \bigg| [/tex]

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

    [tex] \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}} [/tex]

    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.
     
  7. Jul 17, 2015 #6
    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?
     
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