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
- multiplication/division
- raising to powers and roots
- trigonometric functions
- exponential functions
- logarithmic functions
, which when differentiated gives the function $f(x)$. I've heard this said about the function $f(x) = x^x$, for example.
What sort of techniques are used to prove statements like this? What is this branch of mathematics called?
Merged with "How to prove that some functions don't have a primitive" by Ismael:
Sometimes we are told that some functions like $\dfrac{\sin(x)}{x}$ don't have an indefinite integral, or that it can't be expressed in term of other simple functions.
I wonder how we can prove that kind of assertion?
Answer
It is a theorem of Liouville, reproven later with purely algebraic methods, that for rational functions $f$ and $g$, $g$ non-constant, the antiderivative
$$f(x)\exp(g(x)) \, \mathrm dx$$
can be expressed in terms of elementary functions if and only if there exists some rational function $h$ such that it is a solution to the differential equation:
$$f = h' + hg$$
$e^{x^2}$ is another classic example of such a function with no elementary antiderivative.
I don't know how much math you've had, but some of this paper might be comprehensible in its broad strokes: http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf
Liouville's original paper:
Liouville, J. "Suite du Mémoire sur la classification des Transcendantes, et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients." J. Math. Pure Appl. 3, 523-546, 1838.
Michael Spivak's book on Calculus also has a section with a discussion of this.
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