Math by Tathagata
The orthogonal and orthonormal functions on an interval [a, b] are such functions where the tangents to the curves y=Φ1(x) and y=Φ2(x) at their intersecting points are perpendicular to each other. In this article, we study orthogonal and orthonormal functions along with examples. Orthogonal Function Definition A set of functions {Φ1(x), Φ2(x), …, Φn(x), …} … Read more
The following are the assignments on special functions you need to submit. The questions are given below: Assignment Problems Due Date: 26th Nov. Q1: Prove that the generating function for Hermite polynomial is $e^{2tx-t^2}$. That is, if $e^{2tx-t^2}=\sum_{n=0}^\infty H_n(x) \dfrac{t^n}{n!}$ then find $H_n(x)$. Q2: Find the values of H2n(0) and H2n+1(0). Q3: Show that $H_n(x)$ … 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 nth derivative of a function f(x) is obtained by by successively differentiating f(x) n times with respect to x. The Leibnitz theorem is a powerful tool to find it for a product function. In this post, we will learn how to find n-th derivative of a function with examples. Definition of nth Derivative The … Read more
The following are the assignments for the upcoming exam on Engineering Mathematics I. Please solve the questions and submit. Some Study Material: ** End Sem Syllabus: Unit I , II, III, and IV. Unit I and IV: Differential Calculus & Vector Algebra Note: This section is for Sec G and Sec L. Q1: Show that … Read more