Titlebook: Elliptic Integrals and Elliptic Functions; Takashi Takebe Textbook 2023 The Editor(s) (if applicable) and The Author(s), under exclusive l

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Applications of Jacobi’s Elliptic FunctionsIn the previous chapter we defined Jacobi’s elliptic function sn as the inverse function of the incomplete elliptic integral of the first kind, introduced cn and dn and studied their properties. These Jacobi’s elliptic functions appear in various problems, from which we pick up two applications to physics in this chapter.

HACK

Elliptic CurvesExercise 7.2 Prove the above proposition. (Hint: (i) follows from the fact that 𝜑(𝑧) does not have multiple roots. For (ii) and (iii) use local coordinates 𝑧 (around a point which is not a branch point) and 𝑤 (around branch points), after checking that they really are local coordinates.)

從屬

Complex Elliptic IntegralsThe expression (8.4) means that ‘in the homology group any closed curve is equivalent to a curve which goes around 𝐴 several times and then goes around 𝐵 several times’. This can be explained in the following way.

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神圣在玷污

Hay-Fever

The Weierstrass ?-FunctionIn the previous chapter we defined elliptic functions as meromorphic functions on an elliptic curve = doubly periodic meromorphic functions on . and studied their properties. In particular, we gave several examples of elliptic functions which are obtained immediately from the definitions.

冒失

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Characterisation by Addition FormulaeIn the previous chapter we proved that rational functions, rational functions of an exponential function and elliptic functions have addition theorems (algebraic addition formulae). Are there other functions which have algebraic addition formulae? The next Weierstrass–Phragmén theorem1 answers this question.

刀鋒

https://doi.org/10.1007/978-3-031-30265-7elliptic functions; elliptic integrals; complex analysis; application to physics; Riemann surfaces; ellip

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