John Bredakis method and highways without speed limit.
Cook book forumlas for integrals.
A serious attempt to make my method and the gamma function accessible to anyone dealing with integral calculus. The bing bang derivation of innumerous mathematical formulas starting from elementary trigonometrie.
Table of Contents
1. A fascinating look at Higher Mathematics
2. John Bredakis method and Highways without speed limit
3. Cook book formulas for the following integrals
4. Specific formulas
5. Improper integrals
6. Laplace Transformation and the relation of some transforms to the gamma function Γ(x) and to the John Bredakis method
7. Instead of providing a table of inverse Laplace Transforms
8. A summary of the gamma Γ(x) and the beta B(x,y) function
9. Contribution of the gamma function Γ(x) to the normal distribution
10. Applications of the beta function B(x,y)
11. Formula of B(x,y) and the Jacobian
12. Formulas and remarks related to the gamma function Γ(x)
13. Various proofs related to the gamma Γ(x) and beta function B(x,y)
14. The proof of the equivalent forms of B(x,y)
15. Just to get an idea: This topic is at very advanced level
16. Summary of the properties of the solutions of the classical Bessel function and the basic integrals of those solutions
17. Bessel's equation plays an important role in mathematical analysis
18. Back to my method
19. The story behind my method
Research Objectives and Core Topics
This work aims to provide a comprehensive, "cook-book" style approach to solving various complex integrals and mathematical transformations by establishing a systematic methodology derived from elementary trigonometry and the properties of the gamma and beta functions. It seeks to bridge the gap between abstract mathematical theory and practical, applicable solution formulas for physicists and engineers.
- Derivation of a unified method for evaluating improper and standard integrals.
- Application of gamma and beta functions within probability distributions and Laplace transformations.
- Systematic analysis of Bessel's equation and its role in orthogonal function expansions.
- Integration of trigonometric properties through complex exponentiation and identity proofs.
Excerpt from the Book
The story behind my method
After many many years of self education in higher mathematics I reached a satisfactory level in various fields of this science.
Integral calculus is my favourite topic
At one time in the past I tried (without success) to find a method to provide cook book formulas for the integrals of: n integral t.cos(k.t).dt and n integral t.sin(k.t).dt for high integer n.(n>2)
Along this direction I tried the integral e^x.x^n.dx and bingo. My sound knowlegde of the the gamma function Γ(x) verified my general formula at the improper form of this integral.
Studying the differential equations I came across the method of undetermined coefficients from the reference 9. I applied this method to the integral e^(r.x).x^n.dx and bingo. Again the verification came from the improper forms of Laplace Transformation.
The joy of discovery cannot be described
Summary of Chapters
A fascinating look at Higher Mathematics: This introductory section establishes the premise of deriving complex mathematical formulas starting from basic trigonometric principles.
John Bredakis method and Highways without speed limit: This chapter introduces the author’s primary methodology for providing standardized formulas for various integrals and improper forms.
Cook book formulas for the following integrals: A collection of practical, ready-to-use derivative and integral formulas involving exponential and logarithmic functions.
Specific formulas: Detailed mathematical identities and formulas designed for specialized integration tasks.
Improper integrals: Focuses on the evaluation of improper integrals and their convergence within specific mathematical domains.
Laplace Transformation and the relation of some transforms to the gamma function Γ(x) and to the John Bredakis method: Discusses the application of the Laplace transformation and its deep connections to gamma function properties.
Instead of providing a table of inverse Laplace Transforms: Offers examples and practical applications for determining inverse transforms without relying on static reference tables.
A summary of the gamma Γ(x) and the beta B(x,y) function: A mathematical overview of the properties, definitions, and identities of gamma and beta functions.
Contribution of the gamma function Γ(x) to the normal distribution: Explores the relationship between the gamma function and fundamental concepts in probability theory, specifically the normal distribution.
Applications of the beta function B(x,y): Provides concrete examples and formulas demonstrating the utility of the beta function in advanced calculus.
Formula of B(x,y) and the Jacobian: Derives relationships using coordinate transformation and the Jacobian determinant.
Formulas and remarks related to the gamma function Γ(x): Compiles Wallis-derived formulas and factorial rules essential for gamma function operations.
Various proofs related to the gamma Γ(x) and beta function B(x,y): A technical collection of proofs validating the fundamental identities of these special functions.
The proof of the equivalent forms of B(x,y): Demonstrates the consistency of beta function representations through rigorous proof.
Just to get an idea: This topic is at very advanced level: An overview of the solutions to Bessel’s equation and their significance in advanced analysis.
Summary of the properties of the solutions of the classical Bessel function and the basic integrals of those solutions: Lists the specific integral forms and properties associated with Bessel function solutions.
Bessel's equation plays an important role in mathematical analysis: Analyzes the conversion of Bessel's equation into Sturm-Liouville form and its utility in coordinate system selection.
Back to my method: Reaffirms the trigonometric foundations of the author's general method through series expansions and complex identities.
The story behind my method: A biographical and intellectual reflection on the author's development of these mathematical techniques.
Keywords
Integral Calculus, Gamma Function, Beta Function, Laplace Transformation, Bessel's Equation, Trigonometry, Higher Mathematics, Improper Integrals, Differential Equations, Wallis Product, Normal Distribution, Mathematical Analysis, Orthogonal Functions, Jacobian, Undetermined Coefficients
Frequently Asked Questions
What is the core focus of this publication?
This work focuses on providing a systematic, "cook-book" style approach for solving complex integrals, primarily utilizing trigonometric foundations and properties of special functions.
What are the primary fields covered?
The publication covers integral calculus, Laplace transforms, special functions (Gamma and Beta), Bessel’s equations, and their applications in mathematical analysis and probability.
What is the author's main objective?
The primary objective is to present a verified methodology for deriving formulas that simplify the resolution of complex improper integrals for practical use in science and engineering.
What mathematical methods are employed?
The author utilizes methods of undetermined coefficients, coordinate transformations, series expansions, and rigorous mathematical proofs derived from fundamental trigonometric identities.
What does the main body of the text contain?
The main body contains exhaustive derivations of integral formulas, property summaries for Gamma, Beta, and Bessel functions, and practical applications for solving differential equations.
Which keywords characterize this work?
The work is characterized by terms such as Integral Calculus, Gamma Function, Beta Function, Laplace Transformation, Bessel’s Equation, and Trigonometry.
How is the Gamma function linked to the normal distribution?
The text demonstrates how the Gamma function serves as a fundamental component in calculating the mathematical expectation and variance within probability distributions.
Why is Bessel’s equation significant in this analysis?
Bessel’s equation is highlighted for its role in Sturm-Liouville form conversions, which are essential for expansion into complete sets of orthogonal functions.
How does the author derive these formulas?
The formulas are derived starting from elementary trigonometry and utilizing the relationship e^(i.p) = cos(p) + i.sin(p) to unify various mathematical properties.
- Citar trabajo
- Prof. Dr. med. John Bredakis (Autor), 2011, A fascinating look at higher Mathematics (expanded version), Múnich, GRIN Verlag, https://www.grin.com/document/170470
