Questions tagged [differential-field]

A differential field is a commutative field equipped with derivations.

A differential field is a commutative field $K$ equipped with derivations which are unary functions that are linear and satisfy the Leibniz product rule.

A natural example of a differential field is the field of rational functions in one variable over the complex numbers, $\mathbb{C}(t)$, where the derivation is differentiation with respect to $t.$

12 questions
156
votes
1 answer

How to determine with certainty that a function has no elementary antiderivative?

Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental functions) such that $ \frac{d}{dx}F(x) = f(x)$? In other…
hesson
  • 1,984
  • 3
  • 16
  • 19
11
votes
0 answers

Is there a solvable differential equation with a nonsolvable lie group of symmetries?

For a polynomial equation in one variable over $\mathbb{Q}$, it is well known that the equation is solvable by radicals if and only if the equation's Galois group (which is a finite group) is solvable. The 'only if' part is important - we need it to…
3
votes
2 answers

How does one make real functions a differentiable field?

If you want to apply the results of differential field theory to actual $\Bbb R\to\Bbb R$ functions, then first of all you have to find operations that make these functions a field. The trouble is that with the standard definition of function…
Jack M
  • 26,283
  • 6
  • 57
  • 113
3
votes
0 answers

When does $\sqrt{f(x)}\exp{g(x)}$ have an elementary antiderivative?

Liouville's original criterion for elementary anti-derivatives states: If $f,g$ are rational, nonconstant functions, then the antiderivative of $f(x)\exp{g(x)}$ can be expressed in terms of elementary functions if and only if there exists a…
2
votes
1 answer

Is a factorial an algebraic function and an elementary function?

Following is a question spun off from a comment I received: is a factorial an elementary function and an algebraic function? From elementary functions by Wikipedia By starting with the field of rational functions, two special types of…
Tim
  • 43,663
  • 43
  • 199
  • 459
2
votes
1 answer

The field of constants of a differential ring. Derivative of real and complex numbers.

Let $D$ be a derivation operator over a ring $R$: $$D(a + b) = D(a) + D(b) \\ D(ab) = D(a)b + aD(b)$$ for all elements $a,b\in R$. If the ring is the field $\mathbb{Q}$, all derivatives should be equal to zero since $$D(0)=D(0+0)=2D(0)=0 \\…
1
vote
0 answers

Canonical reference for algebraic theory of polylogs?

I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to calculate some integral involving a complicated…
1
vote
1 answer

Is this an isolated equilibrium point?

I've just been learning the definition of an isolated equilibrium point. From my understanding of this definition, I would expect (as an example) the point $x=1$ to be an isolated fixed point for the differential equation $\dfrac{\mathrm dx}{\mathrm…
0
votes
1 answer

Does an elementary antiderivative of $e^{\sin x} \sin x$ exist?

I wonder if an elementary antiderivative of the function $e^{\sin x} \sin x$ exist? If so, could anyone help me to derive this certain antiderivative step by step? If not, is a strict proof of the nonexistence available, maybe by using knowledge of…
0
votes
0 answers

German for "Liouvillian extension"

How do I correctly translate "Liouvillian extension" to german, especially "Liouvillian"? "Liouvillsche Erweiterung" sounds rather strange, but might be correct. Anyone knows if this is correct?
0
votes
0 answers

Defining the rational function field in n variables.

Reading over an editing my dissertation "Elementary functions" and i am having trouble with my definition of a rational functions in n variables, this is what i have written but its missing one part: Definition Let F be a differential field…
ENAFMTH
  • 433
  • 2
  • 11
0
votes
2 answers

$\int \cos(x) \ln(x) dx$, elementary function?

My course book bluntly mentions (freely translation without any proof): Integral functions with the terms $x^{\alpha} \sin(\beta x)$, $x^{\alpha} \cos(\beta x)$ or $x^{\alpha}e^{\beta x}$ ($\alpha, \beta\in \mathbb R$) are elementary if $\beta=0$…
hhh
  • 5,323
  • 6
  • 45
  • 98