Math by Tathagata
Tathagata In the post, we will learn about complex differentiation where we study the derivative of functions of a complex variable along with some solved problems. ℂ := The set of complex numbers. Complex Differentiation: Definition Let D ⊆ ℂ be an open set and let f: D→ℂ be a complex function. The function f is … Read more
A tree with n vertices has (n-1) edges. For example, a tree with 5 vertices should have 5-1=4 edges. In this post, we prove that a tree contains (n-1) edges if it has n vertices. A tree of n vertices has n-1 edges Theorem: Show that a tree with n vertices has (n-1) edges. Proof: … Read more
For n≥2, a simple graph with n vertices has a pair of vertices of same degree. In this post, we will prove this theorem on simple graphs. Before we do this, let us first recall a simple graph which is given below. A graph is called a simple graph if there is only one edge … Read more
This is the assignment set on Discrete Mathematics for Sec C. The questions are given as follows: Problems on Discrete Math Due Date: 26th Nov Q1: When a statement are considered to be contingency/contradiction? Q2: What is an equivalence relation? Give an example. Q3: If 5 people are seated about a round table then how … Read more
The study material of Special Functions are given on this page. Special Functions Practice Problems. Some practice problems of Special Functions are given here. Mid Term Practice Problems My Gadgets TathagataThis article is written by Dr. Tathagata Mandal, Ph.D in Mathematics from IISER Pune (Algebraic Number Theory), Postdocs at IIT Kanpur & ISI Kolkata. Currently, … Read more
The Cayley-Hamilton theorem is about the characteristic equation of a square matrix. Using this theorem one can find the inverse of a matrix, the integral power of a matrix, and many more. In this post, we will study this theorem along with some applications. Cayley-Hamilton Theorem Statement Statement: Every square matrix satisfies its own characteristic … Read more
The characteristic equation of a matrix is an important tool to study a matrix. Using this equation, we can find the eigenvalue with its corresponding eigenvectors to analyse a matrix. In this post, we will learn about the characteristic equation of a matrix. Definition Let A be an n×n matrix over a field F. The … Read more
Rolle’s theorem is very useful in Calculus for continuous functions. In this post, we will learn Rolle’s theorem with its geometrical interpretation along with some solved examples. Statement of Rolle’s Theorem Let f(x) be a real-valued function defined on the closed interval [a, b] satisfying the following: Then there is a point c ∈ (a, … Read more
The following questions on Complex Analysis can be treated as an assignment as well as the suggestions on the upcoming exam. Q1: Find the real and the imaginary part of $f(z)=\dfrac{\overline{z}}{z}, z \neq 0$. Q2: Define the differentiability of a function f(z) at z=z0. Prove that f(z) = |z| is not differentiable at z=0. Q3: … Read more
The following problems on Mathematical Analysis can be treated as an assignment for the course MTH275. Please solve the problems (as many as you can) and submit to me. Real Analysis Q1: Prove that √2, √3 are irrational numbers. Q2: State Cauchy’s principle for the convergence of a series. Using this principle show that the … Read more
