Abstract
Suppose ω 1 and ω 2 are complex numbers that satisfy Im(ω 2∕ω 1) > 0, and let
We show how properties of Λ(ω 1, ω 2) and certain of its subsets imply transformation formulas for elliptic functions and modular forms. All positive weights can be handled in the same way, and the conditionally convergent cases of weights one and two present no extra difficulty.
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Notes
- 1.
The function Z(τ, ℓ) being defined in (2.30) is completely different from the function Z(θ, ω 1, ω 2) in (2.7). They can be distinguished from each other because one is a function of two variables whereas the other is a function of three variables. No confusion will arise, because the functions occur in different contexts and do not normally appear together.
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Cooper, S. (2017). Transformations. In: Ramanujan's Theta Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-56172-1_3
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DOI: https://doi.org/10.1007/978-3-319-56172-1_3
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