A collection of some of my notes. Some of them were written for teaching purposes, while others were written for entertainment. In case you find any blunders/errors/typos/grammatical mistakes (however small or big it might be), I would greatly appreciate if you could let me know.

#### Algebra

#### Analysis

Different ways of proving \(\displaystyle\int_{\mathbb{R}} e^{-x^2}dx\)
Evaluating \(\displaystyle \int_0^{\pi/2} \sin^k(t) dt\) or \(\displaystyle \int_0^{\pi/2} \cos^k(t) dt\)
Wallis formula: \(\dfrac{\pi}2 = \displaystyle \prod_{n=1}^{\infty} \left(\dfrac{2n}{2n-1} \cdot \dfrac{2n}{2n+1}\right)\) and asymptotic of the central binomial coefficient
De Moivre's and Stirling's formula
An identity on product of sines
Ostrowski's Theorem
Fresnel integral
\(\displaystyle \int_0^{\pi}\ln(\sin(x))dx = -\pi \ln(2)\)
Frullani Integral
Bernoulli Inequality
Summation of sine and cosine
Evaluating \(\displaystyle \int_0^{\pi/2} \dfrac{dt}{1+r\sin(t)}\) and related integrals
An elegant way to prove that \(\lim_{n \to \infty} n^{1/n}=1\)
An infinite product identity; Solution to SIAM problem posted by Carlo Sanna
Divergence of harmonic series
Completeness of \(\mathbb{R}\)
A new way of regularizing sums of the form \(1^k+2^k+\cdots+n^k+\cdots\)
Motivation for doing real analysis course
Weierstrass approximation theorem
Markov and Chebyshev inequalities
Holder's inequality
Minkowski's inequality
Cantor's intersection theorem
Bolzano Weirstrass Theorem
#### Combinatorics

A note on 2048 game
Scoreline in a 'n' match cricket test series
Inclusion exclusion principle
Constructing a \(3 \times 3\) normal magic square
Josephus Problem
#### Geometry

Gauss Lucas theorem
Baudhayana Apastamba theorem (Neo-historians call it as Pythagoras theorem)
Brahmagupta area formula
Ptolemy's theorem
Bhaskara's proof of Baudhayana-Apastamba Theorem
Isoperimetric inequality
Ceva's and Menelaus' theorem and their converses
Determining center of a circle using only compass (an equivalent version of the Napoleon's problem)
Angle bisector theorem
Stewart theorem
Lengths of median, altitude and angle bisector
Intercept theorem
Classification of quadrilaterals
#### Number Theory

Bezout's Lemma
Pell's equation: Ramanujan and Mahalanobis
Characterization of even perfect numbers
Divisors
Euler Totient function
Euler's theorem (generalized Fermat's little theorem)
Product of \(k\) consecutive integers divisible by \(k!\)
Characterizing Pythagorean triplets
Wilson's Theorem
Lagrange's theorem (Number Theory)
Characterizing Three Squares in Arithmetic Progression
Fermat's Right Triangle Theorem
Some properties of Pythagorean triplets
No four squares are in non-trivial Arithemtic Progression
Numbers as difference of two squares
Thue's Lemma
The harmonic number \(H_n\) is never an integer for \(n > 1\)
Infinitude of the set of primes
Divergence of sum of reciprocal of primes
Goldbach-Euler theorem
Highest power of a prime dividing \(n!\)
Abel summation lemma/Method of partial summation
Euler-Maclaurin summation
Bertrand's postulate
Chebyshev bounds
#### Numerical

Catastrophic round-off in finite arithmetic. The file catastrophic_round_off.m.
Solving \(u_{xx}=f\) numerically
Fundamental theorem of polynomial interpolation
Orthogonal polynomials
Chebyshev polynomials
Legendre polynomials
Maximum eigenvalue ratio doesn't always indicate conditioning of the matrix
Gram-Schmidt QR decomposition
Pivoting in Linear systems
Error in computing integrals recursively
Backward stability of triangular systems
Backward stability of inner and outer products
#### Set Theory

Cantor's theorem
Denumerability of \(\mathbb{N} \times \mathbb{N}\)
#### Differential equations

Poisson's Equation in 1D
Variation of parameters
Adjoint of second order ODE, Lagrange's identity and Green's second identity
Sturm-Liouville operator
Uniqueness for Laplace equation
#### Others

Tautochrone
Projectile