Abstract
This chapter discusses the theta functions as the building blocks of the elliptic functions and describes their behavior under the action of two of the fractional linear transformations groups in addition to the derivation of various identities. These identities are easily deduced from the properties of the theta functions and the simplest facts about elliptic functions. The chapter presents an elliptic function constructed in terms of theta functions and a proof of the coefficients of the Laurent expansion of this function being modular forms. This function is a disguised version of the P-function. The chapter explores the classical formulas for e1 (τ), e2 (τ), and e3 (τ) in terms of the thetas and illustrates that the well-known formula Σ ei = 0 is equivalent to an identity among the thetas. The second and third elementary symmetric functions of e1, e2, and e3 are modular forms of degree two and three, respectively; these modular forms are polynomials in the thetas.
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