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Mar. 1st, 2007 02:04 pmI think I shall be able to explain k6 = (2 - √3)(√3 - √2) without using theta functions! YAY
You must fear THE CUBIC MODULAR EQUATION
AIEEEEEE!
I have to prove that and then copy down an argument from this paper. The significance of the "singular modulus" k6 is that if K is the elliptic integral of the first kind, then
This gets used in the paper to make an expression look nicer. But it's hard to prove that k6 is as above. (I also have to get k2/3 but VERY fortunately the argument above produces both of them at the same time.)
Many formulae for other singular moduli are known ("many" because they all produce different answers!), and there are complicated general techniques to find expressions for them. This was one of Ramanujan's big areas of contribution. (Prof. Berndt's trademark appears to be to put "Ramanujan" in the title of his papers?) Unfortunately in order to talk about this, I would have to understand it; also I would have to break my promise in the introduction not to use theta functions.
You must fear THE CUBIC MODULAR EQUATION
I have to prove that and then copy down an argument from this paper. The significance of the "singular modulus" k6 is that if K is the elliptic integral of the first kind, then
Many formulae for other singular moduli are known ("many" because they all produce different answers!), and there are complicated general techniques to find expressions for them. This was one of Ramanujan's big areas of contribution. (Prof. Berndt's trademark appears to be to put "Ramanujan" in the title of his papers?) Unfortunately in order to talk about this, I would have to understand it; also I would have to break my promise in the introduction not to use theta functions.
