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