Elliptic Functions: Infinite Products

Preliminary Results

Let us consider the ascending Landen sequence of moduli $$\cdots < k_{-n} < k_{-(n - 1)} < \cdots < k_{-2} < k_{-1} < k_{0} = k < k_{1} < k_{2} < \cdots < k_{n} < \cdots$$ where $$k_{n + 1} = \frac{2\sqrt{k_{n}}}{1 + k_{n}},\,\, k_{n} = \frac{1 - k_{n + 1}'}{1 + k_{n + 1}'}$$ Then it can be checked easily that the sequence of complementary moduli in reverse order $$\cdots < k_{n}' < k_{n - 1}' < \cdots < k_{2}' < k_{1}' < k_{0}' = k' < k_{-1}' < k_{-2}' < \cdots < k_{-n}' < \cdots$$ also forms an ascending Landen sequence.

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Elliptic Functions: Landen's Transformation

In our introductory post we had talked about the similarities of the elliptic functions with the circular functions. At the same time we discussed the properties which were quite unlike those of circular functions (like double periodicity). In this post we are going to discuss some further properties of elliptic functions which have no analogue in the theory of circular functions.

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Elliptic Functions: Double Periodicity Contd.

In the previous post, we had some general discussion on the periodicity and came to the conclusion that an analytic function can have at most two independent periods and in that case the ratio of periods can not be real. It was also established that if $ \omega_{1}$ and $ \omega_{2}$ are two independent periods then any period $ \omega$ can be expressed as $ \omega = m\omega_{1} + n\omega_{2}$ where $ m, n$ are integers.

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