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85
votes
6answers
7k views

How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
1
vote
0answers
61 views

May the integral $\int\root 3 \of{\cos(x)^2}\,dx$ be expressible by elementary functions?

I would like to decide by methods of differential algebra whether the integral $\int\root 3 \of{\cos(x)^2}\,dx$ might be contrary to the output of CAS Mathematica Online Integrator expressible by ...
10
votes
2answers
496 views

proof that $e^x$ is a transcendental function of $x$?

Let a function $f(x)$ be algebraic if it satisfies an equation of the form $$c_n(x)(f(x))^n + c_{n-1}(x)(f(x))^{n-1} + \cdots + c_0(x)=0,$$ for $c_k(x)$ rational functions of $x$, and let $f$ be ...
3
votes
0answers
42 views

Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
0
votes
0answers
24 views

The Risch algorithm

I tried to understand the Risch Algorithm and I was able to follow the cases for exponential and logarithmic extensions. However, I could not understand what problem arises in algebraic extension. I ...
2
votes
1answer
45 views

What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I ...
3
votes
1answer
104 views

Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
0
votes
0answers
41 views

How we transform our differential algebraic equations (DAEs) to ordinary differential equations (ODEs)?

Higher index DAEs are difficult, we need to transform it to ODEs to make it easier to solve. My question is how do we transform DAEs to ODEs? I hope somebody can help me, thanks in advance and have a ...
1
vote
1answer
63 views

Controllability on nonlinear systems

Dynamic system in the book (chapter 6, page 67 in http://www.me.berkeley.edu/ME237/6_cont_obs.pdf) $$ \begin{cases} \dot{x}_{1}=x_{2}^{2}\\ \dot{x}_{2}=u\end{cases} $$ so $$ ...
9
votes
1answer
121 views

Theorem: Anti-differentiation is harder than differentiation

The question of why anti-differentiation is "harder" than differentiation was the topic of an earlier question, and some of the answers are interesting, but I'm not sure they fully answer it, and this ...
4
votes
0answers
38 views

If $D:A\to A$ is a derivation, what can be said about the range of $D$?

What can be said about the relation between the domain and range of a derivation as a function? If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? ...
2
votes
1answer
113 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
3
votes
1answer
97 views

Trace and Norm maps on differential extensions

I'm working through a proof which is rather algebraic, and my abstract algebra is probably only basic to intermediate. I have a differential extension $E/K$ of a differential field $K$, and the proof ...
2
votes
1answer
178 views

Introduction to Elementary Functions

I'm looking for an introductory text on algebraic treatment of elementary functions. Really short and easy-going. Video lectures are even better. I want to learn basic ideas (i.e. definitions) behind ...
0
votes
1answer
83 views

“fluent” functions

In an old mathematics book (Ritt, 1948, p.5) I have come across the notion of "monogenic analytic" and "fluent" functions. These are complex valued functions. Has anyone heard of these terms before? ...
0
votes
3answers
90 views

What is $\lim\limits_{x\to 0}x^2\cos(2/x)$?

I have to evaluate $$\lim_{x\to 0}x^2\cos(2/x)$$ using one or more of the limit laws. I am using the multiplication law and I am wondering if I am on the right track here? I have split it up to: ...
1
vote
0answers
67 views

Algebras vs. rings in algebraic differential calculus

Vector fields (and differential operators of higher order) on a real manifold are often defined in terms of $\mathbb{R}$-algebra $C^\infty(M)$. However, it is not clear to me why is the ...
3
votes
3answers
460 views

What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology

For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
4
votes
2answers
279 views

Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...