# Tagged Questions

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14k 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 ...
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### 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 ...
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### Why differential Galois theory is 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. My ...
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### What is a constant field?

I am looking at the following: Could you explain to me what a constant field is?  P.S. I found this in the paper of T. Honda, "Algebraic differential equation" (pages 170-176).
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### What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, Picardâ€“...
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### 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 ...
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### The solutions are linearly independent and algebraic

The Grothendieck problem for differential equations (Grothendieck-Katz conjecture) is the following: $$\alpha_n(x)y^{(n)}(x)+\dots +a_1 (x)y'(x)+a_0(x)y(x)=0, a_i \in \mathbb{Z}[x]\ \ \ \ (*)$$ We ...
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### How can I solve this difference equation, if $p=q$?

I'm trying to solve the difference equation $$pE_{k+1} - E_{k} + qE_{k-1} = -1$$ given the boundary conditions $E_{0} = 0, \; E_{a} = 0$, if $p=q=\frac{1}{2}$ To attempt this, I first found and ...
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### Joint Continuity implies being a Hausdorff space(in the given case)?

If we consider G to be the group of non-singular n by n matrices carrying Zariski topology. Then the joint continuity of the multiplication map from G x G to G would make G Hausdorff.(We consider the ...
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### 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 ...
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### Proof of nonexistence of a solution to an equation in terms of elementary functions?

In a numerical methods class I'm taking, it was claimed that the equation $A = \frac{R^2}{2} \left(\theta - \sin\theta \right)$ cannot be analytically solved for $\theta$. I don't doubt that this is ...
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### 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 $\mathbb{R}$-...