The Magic of Theta Functions: Contd.

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In the previous post we studied some interesting properties of theta functions which were used to relate them to AGM and thereby to elliptic integrals. We will continue to explore further in this direction and start with a remarkable property of theta function $ \theta_{3}(q)$.

The Magic of Theta Functions

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Theta functions were originally introduced by Carl Gustav Jacob Jacobi while studying elliptic functions (which are in turn related to elliptic integrals). These functions are also connected with number theory and they have many interesting properties besides. Since they are related to elliptic integrals and we have seen in a previous post that the elliptic integrals are related to the AGM (arithmetic-geometric mean), it follows that the theta functions are related to the AGM. We will cover these topics in this series of posts and will also mention some number theoretic applications of theta functions.