A Continued Fraction for Error Function by Ramanujan

Today's post is inspired by this question I asked sometime back on MSE. And after some effort (offering a bounty) I received a very good answer from a user on MSE. This question was asked for the first time by Ramanujan in the "Journal of Indian Mathematical Society" 6th issue as Question no. 541, page 79 in the following manner:

Prove that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + \left(\cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}\right) = \sqrt{\frac{\pi e}{2}}\tag{1}$$ It turns out that the first series is intimately connected with the error function given $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,dt\tag{2}$$ In his famous letter to G. H. Hardy, dated 16th January 1913, Ramanujan gave the following continued fraction for the integral used in the definition of error function given above: $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}\tag{3}$$

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Values of Rogers-Ramanujan Continued Fraction: Part 3

Evaluation of $R(e^{-2\pi/5})$

In the last post we established the transformation formula $$\left[\left[\frac{\sqrt{5} + 1}{2}\right]^{5} + R^{5}(e^{-2\alpha})\right]\left[\left[\frac{\sqrt{5} + 1}{2}\right]^{5} + R^{5}(e^{-2\beta})\right] = 5\sqrt{5}\left[\frac{\sqrt{5} + 1}{2}\right]^{5}\tag{1}$$ under the condition $\alpha\beta = \pi^{2}/5$.

If we put $\alpha = \pi$ then $\beta = \pi/5$ and since we already know the value of $R(e^{-2\pi})$ we can use equation $(1)$ to evaluate $R(e^{-2\pi/5})$. But in order to do that we need to calculate $R^{5}(e^{-2\pi})$ first.

We have from an earlier post $$R(e^{-2\pi}) = \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2}$$

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Values of Rogers-Ramanujan Continued Fraction: Part 2

Continuing our journey from the last post we will deduce further properties of the Rogers-Ramanujan Continued Fraction $R(q)$ which will help us to find out further values of $R(q)$. In this connection we first establish an identity concerning powers of $R(q)$.

Identity Concerning $R^{5}(q)$

Using the identity $(3)$ from the last post, Ramanujan established another fundamental property of $R(q)$ namely: $$\frac{1}{R^{5}(q)} - 11 - R^{5}(q) = \frac{f^{6}(-q)}{qf^{6}(-q^{5})}\tag{1}$$

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Values of Rogers-Ramanujan Continued Fraction: Part 1

A Wild Theorem by Ramanujan

In his letter dated 16th January 1913 to G. H. Hardy, Ramanujan presented the following wild theorem: $$\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + \cdots}}}} = \left(\sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2}\right)\sqrt[5]{e^{2\pi}}\tag{1}$$ The theorem looks so strange and surprising, coming out of nowhere that Hardy had to remark: "they must be true because, if they were not true, no one would have had the imagination to invent them." In this post we will prove the above theorem using elementary methods. The proof is essentially the one given by Watson who claimed that probably Ramanujan obtained the result in the same manner.

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Rogers-Ramanujan Identities: A Proof by Ramanujan

Introduction

The history of Rogers-Ramanujan identities is well described in various books and papers. In brief these identities were first discovered and proved by L. J. Rogers in 1894 and then later re-discovered (but not proved) by Ramanujan in 1913. Later in 1919 Ramanujan published a proof. It is this proof which will be described here.

Ramanujan was not used to publishing proofs of many of his discoveries and hence there is a feeling (even now) that his methods were mystical and often inspired by his dreams. However he did publish some proofs and when we study these it becomes at once very clear that Ramanujan possessed proofs of almost all the results he found, but it was just lack of time and resources due to which he did not record the proofs. Unfortunately this is a big loss for mathematics because from the nature of his formulas it seems that his methods were highly efficient and startling at the same time. The proof we present in this post also has the same qualities and I hope the reader will enjoy going through these.

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Irrationality of π(PI): Lambert’s Proof Contd.

Irrationality of Continued Fractions

We have the following results about the irrationality of some continued fractions:
If $ a_{n}, b_{n}$ are positive integers then:
1) The infinite continued fraction $$\frac{b_{2}}{a_{2} +}\,\frac{b_{3}}{a_{3} +}\,\frac{b_{4}}{a_{4} +}\,\frac{b_{5}}{a_{5} +}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n}$ for all values of $ n$ starting from a certain value $ n = n_{0}$.

2) The infinite continued fraction $$\frac{b_{2}}{a_{2} -}\,\frac{b_{3}}{a_{3} -}\,\frac{b_{4}}{a_{4} -}\,\frac{b_{5}}{a_{5} -}\,\cdots $$ converges to an irrational value, provided that $ a_{n} \geq b_{n} + 1$ for all values of $ n$ starting from a certain value $ n = n_{0}$ and the condition $ a_{n} > b_{n} + 1$ must hold for an infinite number of values of $ n$.

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Irrationality of π(PI): Lambert's Proof

Introduction

After mentioning about the Lambert's famous proof of irrationality of $ \pi$ in an earlier post, it is now time to give it to the readers in its entirety. I need to reiterate the fact that being a far more direct proof than the modern proofs of Ivan Niven, it is still highly neglected by modern authors and educators. The idea of the proof is really elementary but based on the concept of continued fractions which are now deleted from the high school mathematical syllabus. Why this topic is now left out is still unclear to me. One reason which I can guess of is that the manipulations of continued fractions are not so simple (compared to those of an infinite product or a series). The visible form of the continued fraction does not give any idea about its value unless we do the calculations.

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Continued fraction expansion of tan(x)

Introduction

One of the most intriguing formulas which I encountered during my 12th grade was the continued fraction expansion of $ \tan x$. It was given in a book on "Numerical Analysis" and was offered as an example of a formula which converges very fast (like something comparable to the Maclaurin series for $ \sin x$ and $ \cos x$). However like the usual practice followed in mathematics textbooks it was offered without any proof or any context. I checked the formula using calculator and was amazed with its speed of convergence. Just 10 terms were enough to give the result with an accuracy up to 10 decimals.

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