Today we will focus on a topic from elementary geometry namely the concept of measurement of angles. The idea of an angle is a simple one in the sense that it is made by two rays emanating from the same point. But the measurement of angles is not that simple as it appears. Many theorems in elementary geometry deal with ideas which involve the concept of measurement of angles but they assume the understanding of this measurement in an implicit fashion.
Ramanujan's Generating Function for Partitions Modulo 7
Ramanujan's Partition Congruences
Based on the empirical analysis of a table of partitions Ramanujan conjectured his famous partition congruences $$\boxed{\begin{align}p(5n + 4)&\equiv 0\pmod{5}\notag\\ p(7n + 5)&\equiv 0\pmod{7}\notag\\ p(11n + 6)&\equiv 0\pmod{11}\notag\end{align}}\tag{1}$$ and gave some of the most beautiful proofs for them (see here). In addition to these proofs he gave the following generating functions for $p(5n + 4), p(7n + 5)$: $$\sum_{n = 0}^{\infty}p(5n + 4)q^{n} = 5\frac{\{(1 - q^{5})(1 - q^{10})(1 - q^{15})\cdots\}^{5}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{6}}\tag{2}$$ and \begin{align}\sum_{n = 0}^{\infty}p(7n + 5)q^{n}&= 7\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{3}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{4}}\notag\\ &\,\,\,\,\,\,\,\,+ 49q\frac{\{(1 - q^{7})(1 - q^{14})(1 - q^{21})\cdots\}^{7}}{\{(1 - q)(1 - q^{2})(1 - q^{3})\cdots\}^{8}}\tag{3}\end{align} We have already established $(2)$ in one of our posts and this post deals with the identity $(3)$ concerning generating function of partitions modulo $7$.
By
Paramanand Singh
Saturday, July 26, 2014
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: $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}\tag{3}$$
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: $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}\tag{3}$$
By
Paramanand Singh
Saturday, June 21, 2014
Theories of Exponential and Logarithmic Functions: Part 3
In this concluding post on the theories of exponential and logarithmic functions we will present the most intuitive and obvious approach to define the expression $a^{b}$ directly without going to the number $e$ and the function $\log x$. This approach stems from the fact that an irrational number can be approximated by rational numbers and we can find as good approximations as we want. The idea is that if $b$ is irrational and $a > 0$ then we have many rational approximations $b', b'', \ldots $ to $b$ and the numbers $a^{b'}, a^{b''}, \ldots$ would be the approximations to the number $a^{b}$ being defined. Inherent in such a procedure is the belief that we can find as good approximations to $a^{b}$ as we want by choosing sufficiently good rational approximations to $b$. Thus we can see that the numbers $$2^{1}, 2^{1.4}, 2^{1.41}, 2^{1.414}, 2^{1.4142}, \ldots$$ are approximations to the number $2^{\sqrt{2}}$.
By
Paramanand Singh
Sunday, May 11, 2014
Theories of Exponential and Logarithmic Functions: Part 2
Exponential Function as a Limit
In the last post we developed the theory of exponential and logarithmic function using the standard approach of defining logarithm as an integral. In this post we will examine various alternative approaches to develop a coherent theory of these functions. We will start with the most common definition of $\exp(x)$ as the limit of a specific sequence. For users of MSE this is the approach outlined in this answer on MSE.
By
Paramanand Singh
Saturday, May 10, 2014
Theories of Exponential and Logarithmic Functions: Part 1
In the past few months I saw a lot of questions on MSE regarding exponential and logarithmic functions. Most students were more used to the idea of defining $e$ by $\lim\limits_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^{n}$ and then defining the exponential function as $e^{x}$. I tried to answer some of these questions and based on the suggestion of a user, I am trying to consolidate my answers into a series of posts here. One thing which I must mention here is that most students do have an intuitive idea of the exponential and logarithmic functions but many lack a sound theoretical foundation. In this series of posts I will provide multiple approaches to develop a theory of exponential and logarithmic functions. We will restrict ourselves to real variables only.
By
Paramanand Singh
Thursday, May 8, 2014
Abel and the Insolvability of the Quintic: Part 4
We now turn to the goal of this series namely to establish the fact that the general polynomial of degree $5$ or higher is not solvable by radicals over its field of coefficients. Here Abel's argument is quite terse and I have not been able to fully comprehend some parts of it. Also proof of some statements are not provided by Abel because it appeared quite obvious to him. We will provide here a proof which is based on Ruffini's arguments and its later simplification by Wantzel.
By
Paramanand Singh
Saturday, January 4, 2014
Abel and the Insolvability of the Quintic: Part 3
The proof for the non-solvability of polynomial equation of degree $5$ (or more) by radicals obviously has to proceed via method of contradiction. Abel therefore assumed that such a solution was possible for a quintic and then figured out the most general form of such a solution. At the same time Abel observed that the radical expressions occurring in such a form must themselves be rational expressions of the roots desired. This was a key part which Abel proved for the first time. This result was later termed as the Theorem of Natural Irrationalities.
By
Paramanand Singh
Friday, January 3, 2014
Abel and the Insolvability of the Quintic: Part 2
In the last post we defined the concept of a radical field extension along the lines of the definition of algebraic functions given by Abel. In the current post we will study some properties of such field extensions which will ultimately enable us to study the field extension $\mathbb{C}(x_{1}, x_{2}, \ldots, x_{n})$ of $\mathbb{C}(s_{1}, s_{2}, \ldots, s_{n})$ where $s_{1}, s_{2}, \ldots, s_{n}$ are elementary symmetric functions of the indeterminates $x_{1}, x_{2}, \ldots, x_{n}$.
By
Paramanand Singh
Thursday, January 2, 2014
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