Questions tagged [differential-field]
A differential field is a commutative field equipped with derivations.
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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 ...
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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.
$$ \mathrm{Si}(x) = \int_0^x\!\frac{\sin t}{t}\,\...
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Does an elementary antiderivative of $e^{\sin x} \sin x$ exist?
I wonder if an elementary antiderivative of the function
$e^{\sin x} \sin x$
exist? If so, could anyone help me to derive this certain antiderivative step by step? If not, is a strict proof of the ...
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Canonical reference for algebraic theory of polylogs?
I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to ...
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The field of constants of a differential ring. Derivative of real and complex numbers.
Let $D$ be a derivation operator over a ring $R$: $$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 ...
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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 ...
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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....
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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?
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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 ...
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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:
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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 ...
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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)...
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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, ...
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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 ...
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