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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 $t.$

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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
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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…
roymend
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Does there exist a finite set of solutions to integrals such that any function composed of elementary functions is integrable?

For indefinite integrals whose solutions cannot express with elementary functions, special functions are often defined, such as those shown below. $$ \mathrm{Si}(x) = \int_0^x\!\frac{\sin t}{t}\,\mathrm{d}t \qquad \mathrm{Li}(x) =…
finlay morrison
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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
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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
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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…
Semiclassical
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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…
Franklin Pezzuti Dyer
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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 \\…
Kubrick
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Effective algorithms for differentially closed fields

There are (many) model-theoretic proofs that $DCF_0$, the theory of differentially closed fields of characteristic zero admits quantifier elimination (see "Model Theory" by Marker, Chapter 4 or "A Course in Model Theory" by Ziegler and Tent, Chapter…
toghrul
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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…
M Smith
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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…
zyy
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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?
red_trumpet
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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
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$\int \cos(x) \ln(x) dx$, 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
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