Math by Tathagata
Tathagata 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
The Cauchy-Riemann equations in polar form are given as follows: ∂u/∂r = (1/r) ∂v/∂θ and ∂u/∂θ = – r ∂v/∂r if f(z)=u(r,θ)+iv(r,θ) is an analytic function. These C-R equations are very useful to test the analyticity of a complex function that can be expressed in polar co-ordinates easily. In this article, we learn the polar … Read more
The Cauchy Riemann equation states that if a complex function f(z) is differentiable at z=z0, then ifx(z0) = fy(z0). This is one of the main tool to test whether a complex function is differentiable or not. Here we state and prove Cauchy-Riemann equations along with some solved examples as an application. Statement of Cauchy-Riemann Equations … Read more
The complex analytic functions are one of the main objects to study in Complex Analysis. Here we learn the definition of analytic functions with examples, properties, and solved problems. Key concept to study analytic functions is complex differentiation. ℂ: = The set of complex numbers. Definition of Analytic Function A function f is called an … Read more
