Liouville’s impossibility theorem for elementary integration

Suresh Nayak (ISI Bangalore)

Speaker	Suresh Nayak (ISI Bangalore)
When	Mar 27, 2025 from 09:00 AM to 12:30 PM
Where	LH-111, First Floor
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RUNAWAY SEMINAR

These are seminars (rather discussions) meant for the benefit of the students, where minimal pre-requisites are assumed (in the case of this talk — see below), questions are especially encouraged and there is no real time limit to the discussions. This time we have Prof. Suresh Nayak who will give an exposition of the oft-quoted but seldomly proved theorem of Liouville.

Title: Liouville’s impossibility theorem for elementary integration

Speaker: Suresh Nayak (ISI Bangalore)

Abstract: By a theorem of Liouville, indefinite integrals of functions such as $e^{-x^2}$ or $e^{x}/x$ cannot be expressed in elementary terms, i.e., cannot be expressed in terms of functions built from familiar ones such as the trigonometric functions, their inverses, exponential functions, logarithmic functions etc., using compositions and the usual algebraic operations. I will discuss an algebraic proof of this result which gives some differential criterion for the integral of a function to be expressive in elementary terms.

The talk will not assume much background except for the basic theory of field extensions.