Archives for Mathematical Physics
Abelain Group
Z(p∞) = { z ∈ ℂ | zpk = 1 for some integer k ≥ 1 } Proof that Z(p∞) is an Abelian Group We define the set: Z(p∞) =…
Examples of Taylor SEries versus Fourier Series
Intro Which works better for a given function - a Taylor expansion or a Fourier Expansion? This post explores the pros and cons of each, using specific examples. Examples of…
Taylor Series versus Fourier Series for a function
. Domain of Representation Taylor Series: Works best for local approximations around a single point (Maclaurin series if centered at zero). Fourier Series: Represents a function over an entire interval…
Functions ONLY definable by their integrals – with applications
Functions ONLY Defined by Their Integrals 1. The Gamma Function \( \Gamma(x) \) \( \Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt \), for \( x > 0 \). Applications: Generalization…
Functions Defined by Their Integrals – with applications
Functions Defined by Their Integrals Functions that are defined by their integrals often arise in fields like physics, probability theory, and engineering, where the direct formulation of a function may…
Finite Abelian Groups and Applications to Quantum Physics
Finite Abelian Groups and Applications to Quantum Physics What Are Finite Abelian Groups? A finite abelian group is a group \( G \) with the following properties: Closure: For any…
Galois Groups and Applications to Quantum Mechanics
Finite Galois Groups and Applications in Quantum Mechanics Examples of Finite Galois Groups 1. The Cyclic Group \( C_n \) Example: Consider the extension \( \mathbb{Q}(\alpha_n)/\mathbb{Q} \), where \( \alpha_n…
Borel Algebras and Applications in Physics
Borel Algebra and Applications in Physics Borel Algebra and Applications in Physics Examples of Borel Algebras Real Line (\( \mathbb{R} \)): The Borel algebra on \( \mathbb{R} \) is generated…
Analytic Functions on a Punctured Disk with Applications to Quantum Mechanics
Analytic Functions in Quantum Mechanics and Quantum Field Theory The following examples illustrate how different analytic functions defined on a punctured disk can be applied in quantum mechanics (QM) and…
Quantum Mechanics and the Transformation 1\(z-a)
Quantum Mechanics and the Transformation \( \frac{1}{z - a} \) Transformations of the form \( \frac{1}{z - a} \), especially in the context of complex analysis, appear in quantum mechanics,…