Math by Tathagata
A half range Fourier series of a function f(x) is a Fourier series of f(x) in the interval [0, L] although the function is defined for [-L, L]. Here, we learn about half range Fourier series with examples. Previously, we have learnt about the Fourier series representation of a periodic function of period 2L or … Read more
The Fourier Series of even and odd functions have a nice description. Recall that the Fourier series expansion of a function f(x) with period 2π is given by f(x) = a0 + ∑n=1[ ancos(nx) + bn sin(nx) ] where a0, an, bn are the Fourier coefficients of f(x) (provided that the sum exists). In this article, we learn … Read more
The Fourier series of a periodic function with period P is determined based on the following properties of integrals: $\int_a^{a+P} f(x) ~ dx = \int_0^p f(x) ~dx$. This means the area under the curve y=f(x) over any interval of length P is always the same. Using this property, one can express a function of period … Read more
A Fourier series of a function f(x) with period 2π is an infinite trigonometric series given by f(x) = a0 + ∑n=1[ ancos(nx) + bn sin(nx) ] if it exists. The constants a0, an, bn are called Fourier coefficients of f(x). In this article, let us learn Fourier series along with its formula and examples. … Read more
The periodic functions are such functions whose values are repeated at regular intervals or periods. The period of a periodic function is its main characteristic. In this article, we study the periodic functions with their examples and properties. Periodic Function Definition A function f(x) is called a periodic function if f(x) is defined for all … Read more
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 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
