Extending Our Reach Through Periodic Functions: The Weierstrass ℘-function and the transcendence of \( \frac{{\Gamma {{(1/4)}^2}}}{{\sqrt \pi }} \)
  • Edward B. Burger
  • Robert Tubbs


In this penultimate chapter we look beyond transcendental numbers associated with the familiar function f(z) = e z and examine values associated with a function that has taken center stage in modern number theory—the Weierstrass elliptic function, denoted by ℘ (z). Here we will not only introduce the function ℘(z) and apply it to obtain an attractive transcendence result, but through that extended development we endeavor to highlight the function’s importance. The scope of this development involves both algebraic and analytic aspects of ℘(z). Toward these ends, our approach will be somewhat more relaxed than in the previous chapters. While some readers may classify our treatment here, at moments, as “sketchy,” we would argue that we provide sufficient depth so that the casual reader will be able to appreciate the beauty of the theory, while the motivated reader will be able to fill in any details we may have suppressed.


Entire Function Meromorphic Function Elliptic Curve Identity Element Group Homomorphism 
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Copyright information

© E.B. Burger and R. Tubbs 2004

Authors and Affiliations

  • Edward B. Burger
    • 1
  • Robert Tubbs
    • 2
  1. 1.Department of MathematicsWilliams CollegeWilliamstownUSA
  2. 2.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA

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