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

In summary, Differential Galois Theory states that the function exp(-x^2) does not have an elementary antiderivative, meaning it cannot be expressed in terms of elementary functions such as polynomials, trigonometric functions, and exponentials. This was proven by mathematician Joseph Liouville in the 19th century, and has since been a fundamental result in the theory of differential equations. This means that the integral of exp(-x^2) cannot be evaluated using standard integration techniques, making it a challenging and interesting problem for mathematicians.
  • #1
kowalski
14
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.
 
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  • #3
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.
 

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

1. What is Differential Galois Theory?

Differential Galois Theory is a branch of mathematics that studies the differential equations and their solutions using the tools of Galois theory. It provides a way to determine whether a given differential equation has solutions that can be expressed using elementary functions or not.

2. What is an elementary antiderivative?

An elementary antiderivative is a function that, when differentiated, gives the original function. It can be expressed using standard mathematical operations, constants, and a finite number of elementary functions such as polynomials, exponential, trigonometric, and logarithmic functions.

3. Why does exp(-x^2) have no elementary antiderivative?

This is because exp(-x^2), also known as the Gaussian function, is a special type of function called a "transcendental function". These functions cannot be expressed using a finite number of elementary functions and, therefore, do not have elementary antiderivatives.

4. Can Differential Galois Theory be used to find antiderivatives of other functions?

Yes, Differential Galois Theory can be used to determine whether a given function has an elementary antiderivative or not. If it is proven that the function does not have an elementary antiderivative, then other techniques such as numerical integration or the use of special functions may be needed to find the antiderivative.

5. What are some practical applications of Differential Galois Theory?

Differential Galois Theory has applications in various fields such as physics, engineering, and computer science. It can be used to solve differential equations that arise in these fields and determine whether their solutions can be expressed using elementary functions. It also has applications in cryptography, as it can be used to prove the security of certain encryption algorithms.

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