Differential Galois Theory: exp(-x^2) has no elementary antiderivative

Click For Summary
SUMMARY

Differential Galois Theory provides a systematic approach to determine whether certain elementary functions, such as exp(-x^2) and (sin x)/x, possess elementary antiderivatives. The theory employs algorithmic methods to analyze the solvability of differential equations associated with these functions. Key references and papers can be found on the North Carolina State University website, which offers valuable insights into the application of Galois theory in this context.

PREREQUISITES
  • Understanding of elementary functions and their properties
  • Familiarity with differential equations
  • Basic knowledge of Galois theory
  • Experience with mathematical proofs and algorithms
NEXT STEPS
  • Study the algorithmic methods in Differential Galois Theory
  • Explore the papers available at http://www4.ncsu.edu/~singer/ms_papers.html
  • Learn about the relationship between differential equations and Galois theory
  • Investigate specific examples of functions without elementary antiderivatives
USEFUL FOR

Mathematicians, students of advanced calculus, and researchers interested in the intersection of differential equations and Galois theory will benefit from this discussion.

kowalski
Messages
14
Reaction score
0
A lot of apparently innocent elementary functions, like exp(-x^2) or (sin x)/x, have not antiderivatives in terms of elementary functions. I've read that "Differential Galois theory" explains this, and gives an algorithmic method to know if a given elementary function has or has not elementary antiderivative.
Please, can you explain to me the fundamental, core ideas of this theory?. Some practical, as elementary as possible references? Examples of its use?. Thank you, kowalski.
 
Physics news on Phys.org
Hola Cygni,
thanks for the references; if I learn how to apply effectively this theory (which is my goal) I will post a resume. Thanx again.
 

Similar threads

Undergrad Why is the integral of ##\arcsin(\sin(x))## so divisive?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Matrix Exponential to Approximate the Value of Matrizant
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Runge-Kutta 4 w/ some sugar on the top: How to do error approximation?
  • · Replies 65 ·
3
Replies
65
Views
8K
Undergrad Closed implies exact (differential 1-forms on R^2)
  • · Replies 6 ·
Replies
6
Views
4K
Graduate Triple Product in Laplace Transform
  • · Replies 2 ·
Replies
2
Views
2K
Calculus Differential Equations book recommendations
  • · Replies 5 ·
Replies
5
Views
5K
How can I integrate 2x^2*exp(x^3)?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Deriving the FEA formulation using triangular elements
  • · Replies 2 ·
Replies
2
Views
3K
High Energy Recommended books in HEP, QFT, QM, GR
  • · Replies 9 ·
Replies
9
Views
5K
Graduate Sidney Coleman's opinion on interpretation in his Dirac lecture
  • · Replies 139 ·
5
Replies
139
Views
3K