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WindowTitle->30. Elliptic Functions | Essentials of Complex Analysis
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Elliptic functions are doubly periodic complex functions. Historically, they \
were introduced for the study of elliptic integrals, to which they are \
related.\
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Elliptic integrals were first studied by Euler and Legendre, elliptic \
functions by Abel, Jacobi and Weierstrass.\
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Before defining the elliptic integrals, let\[CloseCurlyQuote]s consider the \
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These two integrals are not elliptic integrals, but have some features in \
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They cannot be expressed as polynomials or roots; instead, \
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By convention, the fundamental domain does not include the \
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An elliptic function without poles in the fundamental domain must be a \
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The sum of residues of all poles in the fundamental domain is zero.\
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The number of zeros and the number of poles in the fundamental domain are the \
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because the function has the same value on two opposite sides of the \
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Nonconstant elliptic functions of order zero can\[CloseCurlyQuote]t exist by \
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Proof of 6: First, it can be shown that for any meromorphic function\
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Elliptic functions of order two, i.e. with two poles in the fundamental \
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Is it possible to use these ideas to define an elliptic function of order 2? \
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The Weierstrass \[WeierstrassP]-function has the following properties, not \
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