Questions tagged [differential-algebra]
The differential-algebra tag has no usage guidance.
10 questions
16
votes
6
answers
1k
views
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f_1(x) = f(x),\ f_{n+...
- 7,127
69
votes
2
answers
25k
views
Does there exist a complete implementation of the Risch algorithm?
Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative?
The Wikipedia article ...
- 82.7k
6
votes
0
answers
867
views
How to extend Ritt's theorem on elementary invertible bijective elementary functions?
The elementary functions according to Liouville and Ritt are the functions of a complex variable built up by applying exponentiation, logarithms and/or algebraic operations finitely often. That means, ...
- 1,053
85
votes
2
answers
20k
views
Why is differential Galois theory not widely used?
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
- 8,071
25
votes
2
answers
2k
views
When can an invertible function be inverted in closed form?
The Risch algorithm answers the question:
"When can a function be integrated in closed form?", see:
https://en.wikipedia.org/wiki/Symbolic_integration
Is anyone aware of any work that answers the ...
- 351
34
votes
2
answers
2k
views
Is it always possible to calculate the limit of an elementary function?
I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
- 496
23
votes
1
answer
1k
views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
- 231
15
votes
2
answers
1k
views
Why do we need admissible isomorphisms for differential Galois theory?
Background: In Kaplansky's Introduction to Differential Algebra, an isomorphism between differential fields $K, L$ is defined to be admissible if $K,L$ are contained in a larger differential field $M$....
- 25.6k
15
votes
2
answers
2k
views
Solvability in differential Galois theory
It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
- 1,214
7
votes
2
answers
815
views
Differential ideal membership problem
We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case?
To be ...
- 1,528