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Ramanujan's Theta Functions pp 243–293Cite as

Levels 1, 2, 3, and 4: Jacobi’s Inversion Theorem and Ramanujan’s Alternative Theories

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Abstract

Jacobi’s inversion theorem states that if 0 < q < 1 and

$$x = 16q\prod _{j=1}^{\infty } \frac{(1 + q^{2j})^{8}} {(1 + q^{2j-1})^{8}},$$

then

$$q = \exp {\rm{ }}\left( { - \pi \frac{{{\;_2}{F_1}\left( {\frac{1}{2},\frac{1}{2};1;1 - x} \right)}}{{_2{F_1}\left( {\frac{1}{2},\frac{1}{2};1;x} \right)}}} \right).$$

We prove this result and provide a similar analysis for three analogous results of Ramanujan.

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Authors and Affiliations

  1. Institute of Natural and Mathematical Science, Massey University, Auckland, New Zealand

    Shaun Cooper

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  1. Shaun Cooper
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Cooper, S. (2017). Levels 1, 2, 3, and 4: Jacobi’s Inversion Theorem and Ramanujan’s Alternative Theories. In: Ramanujan's Theta Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-56172-1_5

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