Jacobi’s Elliptic Functions

  • Derek F. Lawden
Part of the Applied Mathematical Sciences book series (AMS, volume 80)


The elliptic functions sn u, cn u, and dn u are defined as ratios of theta functions as below:
$$sn\;u = \frac{{{\theta _3}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _1}(z)}}{{{\theta _4}(z)}},$$
$$cn\,u = \frac{{{\theta _4}(0)}}{{{\theta _2}(0)}}\cdot \frac{{{\theta _2}(z)}}{{{\theta _4}(z)}},$$
$$dn\;u = \frac{{{\theta _4}(0)}}{{{\theta _3}(0)}}\cdot \frac{{{\theta _3}(z)}}{{{\theta _4}(z)}},$$
where z = u 3 2 (0). sn u is read as “es en yew” or as “san yew”; cn u and dn u can similarly be read letter by letter or as “can u” and “dan u” respectively.


Elliptic Function Theta Function Multivalued Function Simple Pole Fundamental Period 


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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Derek F. Lawden
    • 1
  1. 1.University of Aston in BirminghamBirminghamUK

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