Math by Tathagata
Tathagata For a sequence {an}, if we add all the members of that sequence, then we get an infinite series. Here, a1+a2+a3+… is called infinite series (simply, a series) generated by the sequence {an}. In this article, we will study the convergence of a series along with its definition, formula, examples. The above series is denoted … Read more
The Fourier series of mod x in [-π, π] is given by |x| = π/2 -4/π [cosx/12 + cos3x/32 + cos5x/52 + …]. In this article, we learn how to find the Fourier series of mod x. Fourier Series of |x| Question: Find the Fourier series representation of f(x) = |x| in -π<x<π. Answer: The … Read more
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 even and odd functions are classified by their symmetry properties. A function f is even if f(-x)=f(x) and odd if f(-x)=-f(x). In this article, we study even and odd functions with their properties and solve some problems. Even and odd functions are defined as follows: Even Function: f(-x) = f(x).Odd Function: f(-x) = -f(x). … Read more
The well ordering principle of natural numbers states that every non-empty subset of natural numbers has a least element. Here we state and prove the well-ordering principle with applications. ℕ: = The set of natural numbers. Statement of Well Ordering Principle Every non-empty subset of natural numbers has a least element. That is, if S … Read more