Math by Tathagata
Tathagata In Complex Analysis, zeroes are points where the function vanishes while singularities are points where the function loses its analytic property (differentiability). Here we study zeros and singularities along with their types, examples, residues and related theorems. Zero of a Function Definition: Let f(z): D → ℂ be a function. A point z=a is called … Read more
In graph theory, the degree of a vertex is the number of edges connected to it. In this article, we will study the degree of a vertex in a graph with its definition, examples, and related theorems, such as Handshaking lemma. Degree of a Vertex Definition: In an undirected graph, the number of edges connected … Read more
In this page, we will show that a simple graph with n vertices and k components have at most (n-k)(n-k+1)/2 edges. Question: Prove that a simple graph of $n$ vertices and $k$ components can have at most $\dfrac{(n-k)(n-k+1)}{2}$ number of edges. Answer: Let the $i$-th component of the graph G is denoted by Gi. Suppose … Read more
The sequences (-1)n and (-1)n/n are special type of sequences. In this page, we will study the convergence of the sequences (-1)^n and (-1)^n/n. Convergence of (-1)n Question: Discuss the convergence of the sequence {(-1)n}. Answer: The sequence {(-1)n} can be written as follows: {(-1)n} = {-1, 1, -1, 1, -1, 1, …}. Therefore, the … Read more
To find the residue at pole it is very important to know the order of pole. In this page, we will learn when a singular point can be a pole and how to find the order of that pole. Definition of a Pole A point z=a is called a pole of a complex function f(z) … Read more
In this page, we list Real Analysis practice problems that can be treated as a sample question for the upcoming final exams. Real Analysis Practice Problems Notation: Q1: Define a rational number. Show $\sqrt{3}$ is an irrational number. Q2: Prove that $\mathbb{N}$ has no limit point. Q3: Define an enumerable set. Discuss the enumerability of … Read more
An orthogonal trajectory is a curve that intersects another family of curves at right angles. In this post, we study orthogonal trajectory along with a few questions and answers. Definition of Orthogonal Trajectory An orthogonal trajectory of a family of curves is a curve that intersects every member of that family at a right angle. … Read more
This is the page for Differential Equation practice problems. Here, a list of practice problems will be given for the course on Differential Equation (Exercise). Study Material: (more will follow soon) Ordinary Differential Equation (ODE) Problems Q1: Find the order and degree of $\left(\dfrac{d^3y}{dx^3} \right)^5+x^2y \left( \dfrac{d^4y}{dx^4} \right)^2=\sin(\ln x)$. Q2: Find the value of $m$ … Read more
Here, you can find a list of engineering mathematics questions that can be treated as practice problems in end-sem exam. For the study material of Engineering Mathematics II, please visit this LINK. Engineering Mathematics II Assignments The students of Sections D, G and J need to submit the below assignments on sequence, series and complex … Read more
The particular integral of a differential equation is needed to find to solve the equation. In this article, lets learn how to find particular integral with formula and method. Method of Finding Particular Integral (PI) Let us consider a second order linear differential equation with constant coefficients: $\dfrac{d^2y}{dx^2}+a\dfrac{dy}{dx}+by=Q(x)$ …(1) where a and b constants and … Read more
