Math by Tathagata
Tathagata 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
In the post, we will learn about complex differentiation where we study the derivative of functions of a complex variable along with some solved problems. ℂ := The set of complex numbers. Complex Differentiation: Definition Let D ⊆ ℂ be an open set and let f: D→ℂ be a complex function. The function f is … Read more
A tree with n vertices has (n-1) edges. For example, a tree with 5 vertices should have 5-1=4 edges. In this post, we prove that a tree contains (n-1) edges if it has n vertices. A tree of n vertices has n-1 edges Theorem: Show that a tree with n vertices has (n-1) edges. Proof: … Read more
For n≥2, a simple graph with n vertices has a pair of vertices of same degree. In this post, we will prove this theorem on simple graphs. Before we do this, let us first recall a simple graph which is given below. A graph is called a simple graph if there is only one edge … Read more
In this page, you will find practice problems on Discrete Mathematics. Mid Sem Practice Problems Q1: What is tautology, contradiction and contingency? Give examples. Q2: What is a well-formed formula? (Answer: A well-formed formula is an expression consisting of variables, parentheses, and connective symbols. For example, p∨q.) Q2: Define an equivalence relation on a non-empty … 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 2026 Q1: Define beta and gamma functions. Q2: What are the value of $\Gamma(6)$ and $B(4, 5)$? Q3: Prove that $\Gamma(n+1)=n \Gamma(n)$, and hence prove $\Gamma(n+1)=n!$ Q4: … 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