tag:blogger.com,1999:blog-9004452969563620551.post8058819171828720989..comments2019-05-15T12:04:42.529+05:30Comments on Paramanand's Math Notes: Modular Equations and Approximations to π(PI): Part 1Unknownnoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-9004452969563620551.post-76922343217883977482014-09-08T09:35:03.807+05:302014-09-08T09:35:03.807+05:30@Anonymous
Your formula can be expressed as $$\phi...@Anonymous<br />Your formula can be expressed as $$\phi = e^{\pi/6}\prod_{k = 1}^{\infty}\frac{1 + e^{-5(2k - 1)\pi}}{1 + e^{-(2k - 1)\pi}} = \frac{2^{-1/4}e^{5\pi/24}}{2^{-1/4}e^{\pi/24}}\prod_{k = 1}^{\infty}\frac{1 + e^{-5(2k - 1)\pi}}{1 + e^{-(2k - 1)\pi}}$$ Clearly we can see that the rightmost expression is $G_{25}/G_{1}$. Noting that $G_{25} = \phi$ and $G_{1} = 1$ we establish your formula.<br /><br />For the proof of values of $G_{1}, G_{25}$ read my previous post http://paramanands.blogspot.com/2012/03/ramanujans-class-invariants.htmlParamanand Singhhttps://www.blogger.com/profile/03855838138519730072noreply@blogger.comtag:blogger.com,1999:blog-9004452969563620551.post-60730302928336332732014-09-08T09:24:42.481+05:302014-09-08T09:24:42.481+05:30@Anonymous
In your formula for $\phi$ the product ...@Anonymous<br />In your formula for $\phi$ the product runs from $k = 1$ to $k = \infty$?Paramanand Singhhttps://www.blogger.com/profile/03855838138519730072noreply@blogger.comtag:blogger.com,1999:blog-9004452969563620551.post-74881441182020380102014-09-07T13:06:25.406+05:302014-09-07T13:06:25.406+05:30Here is a similar infinite product series which I ...Here is a similar infinite product series which I found by chance:<br /><br />phi = e^pi/6 * [PI (1+e^-5(2k-1)pi)] / [PI (1+e^-(2k-1)pi)]<br /><br />Of note: The infinite product series (which adjusts the logarithmic spiral at 30 degrees) has as its repeating term: (1 + e^-k(pi)), with k representing the sequence of all odd integers excluding those which are divisible by five (1,3,7,9,11,13,17,19,21,23,27,....).<br /><br />Further, as you show above with the formula for Gn, the same repeating term with k representing all odd integers adjusts the logarithmic spiral at 7.5 degrees to the fourth root of 2.<br /><br />And further, the same repeating term with k representing only the odd integers that are divisible by 5, adjusts the logarithmic spiral at 37.5 degrees to the value of the product of the golden ratio with the fourth root of 2. In this case, the first term of the series is so small that it is easy to see the close relation (at e^5pi/24) with a calculator.<br /><br />I asked the following question on "mathematics.stack.exchange" question: Would this formula fit within, or be obtainable by, the theory of modular units (by Kubert and Lang)? I am entirely unfamiliar with that level of math, but I was told that the theory produces algebraic values for special infinite series products at CM-points. There was no response to the question.Anonymousnoreply@blogger.com