Primary source for this talk: http://www.math.niu.edu/˜ rusin/known-math/93 back/elementary.int, posting by Matthew Wiener to sci.math on Feb 21, 1995 These notes will be available at http://www.stanford.edu/˜ hwatheod Differentiation: mechanical process

• product rule, quotient rule, chain rule

• If we know how to differentiate f_{1}, …, f_{n}, we know how to differentiate any rational function of f_{1}, …, f_{n} and any compositions of f_{1}, …, f_{n}

Integration: not so mechanical

• No product or quotient rule

• ∫ f could involve functions more complicated than f, e.g., integral of a rational function may involve logarithms

• What is ∫ e^{x^{2}} or ∫ \sqrt{tan x}?

Elementary functions An elementary function is any function that can be obtained by starting with the rational functions over C and adjoining a finite number of algebraic elements (radicals, etc.), exponentials, and logarithms. Basic questions:

• How much more complicated can ∫ f be compared to f?

• If f is elementary, is ∫ f also elementary ?

• If ∫ f is in fact elementary, how much more complicated can ∫ f be compared to f ?

Suppose we believe that ∫ f is elementary. How can we find it?

• If we have to adjoin exponentials or logs, they better not be there after we differentiate

• It doesn’t help to adjoin exponentials, since they will survive the differentiaton

• You can adjoin logs, but you can only use them linearly as in c log f(z); if you use them any other way ((log f(z))^{2}, z log f(z), etc.) they will also survive the differentiation Differential fields

A differential field F is a field of characteristic zero with a unary operator ′ : F → F (called differentiation) satisfying the relations

(a + b)′ = a′ + b′
(ab)′ = a′b + ab′

Con(F) = {x ∈ F : x′ = 0}; this is a subfield of F, the constant functions of F. Rational numbers are always constant functions of F.

• usual product and quotient rules

• chain rule: If f ∈ F (x) and a ∈ F, then [f(a)]′ = f′(a) · a′.

• logarithmic differentiation:

\frac{(a_{1}a_{2} … a_{n})′}{a_{1}a_{2} … a_{n}} = \frac{a′_{1}}{a_{1}} + \frac{a′_{2}}{a_{2}} + … + \frac{a′_{n}}{a_{n}}

A simple differential extension of F can be of 3 types. A differential extension of F is obtained by a finite sequence of simple extensions.

(finite extensions) Lemma 1: If K is an algebraic extension of F, then the differentiation operator on F can be extended uniquely to a differentiation operator on K.

(logarithmic extensions) For any 0 ≠ s ∈ F, we adjoin a transcendental element t, with t′ = s′/s, and extend to all rational functions in F (t). (Think ”t = log s”)

(exponential extensions) For any s ∈ F, we adjoin a transcendental element t, with t′ = ts′, and extend to all rational functions in F (t). (Think ”t = e^{s}”)

Lemma 2: If K = F (t) is a logarithmic extension of F, f ∈ F [x], and a is the leading coefficient of f, then

• if a ∉ Con(F), then f(t)′ has the same degree as f(t);

• if a ∈ Con(F), then f(t)′ has one degree less than f(t).

Lemma 3: If K = F (t) is an exponential extension of F, then for any a ∈ F, (at^{n})′ has the form ht^{n} for some h ∈ F. If Con(K) = Con(F) (or at most an algebraic extension) and a ≠ 0, then h ≠ 0. In that case, for any f ∈ F [x], f(t)′ has the same degree as f(t), and is a multiple of f(t) iff f is a monomial.

MAIN THEOREM (Liouville, 1833; present proof due to Rosenlicht, 1968)

Let F be a differential field, a ∈ F. Suppose a is integrable in a differential extension K of F, i.e., there exists y ∈ K with y′ = a, and assume Con(K) = Con(F). Then there exist c_{1}, …, c_{n} ∈ Con(K) = Con(F), and u_{1}, …, u_{n}, v ∈ F, such that

a = c_{1} \frac{u′_{1}}{u_{1}} + c_{2} \frac{u′_{2}}{u_{2}} + … + c_{n} \frac{u′_{n}}{u_{n}} + v′

Then ∫ a = c_{1} log u_{1} + … + c_{n} log u_{n} + v.

(If Con(K) is at most an algebraic extension of Con(F), the result still holds, except that v, the u_{i}’s and the c_{i}’s might lie in an algebraic extension of F.)

In other words, the integral of a must be the sum of a function already in F, plus terms of the form c log u, as we observed at the beginning of the talk.

Lemma 4: (all f’s and g’s are polynomials in some field, f’s nonzero)

(a) If ∑ f_{i}/g^{n_{i}} = 0 with g irreducible and (f_{i}, g) = 1 then there are at least 2 maximal n_{i}’s;

(b) If S_{i} are sums of the form on the left side of part (a) corresponding to distinct g_{i}’s, and ∑ S_{i} = 0 then at least 2 n’s within each S_{i} having nonconstant g_{i} are maximal;

(c) If denominators are not all powers of irreducible polynomials, break into partial fractions and then apply (b);

(d) If ∑ f_{i} = 0, then there at least 2 f_{i}’s of maximal degree.

Examples: ∫ fe^{g}, ∫ e^{z^{2}}, ∫ (z^{2} + az + b)e^{z}/(z − 1)^{2} (from the references)

Liouville’s Theorem (1835) If f, g are rational functions (over C) and ∫ fe^{g} is elementary, then ∫ fe^{g} = we^{g} for some rational function w.

Other results

Theorem (Risch, 1969). There is an algorithm for determining whether an elementary function has an elementary integral.

(Added in October 18, 1999: Risch’s algorithm only covered purely transcendental extensions. Algorithms for the algebraic case were later developed by Trager, Bronstein, and others.)

Theorem (Risch, 1976). If an elementary function has an implicitly elementary integral, it has an elementary integral.

References

1. http://www.math.niu.edu/r̃usin/known-math/93 back/elementary.int, posting by Matthew Wiener to sci.math on Feb 21, 1995

2. D. Mead, Integration, American Mathematical Monthly, Feb. 1961, pp. 152-156.

3. A. Fitt and G. Hoare, The closed-form integration of arbitrary functions, Mathematical Gazette (1993), pp. 227-236.

∫ \sqrt{tan x} dx = \frac{1}{\sqrt{2}} (arctan (\frac{\sqrt{2 tan x}}{1 − tan x}) − log (\frac{1 + tan x + \sqrt{2 tan x}}{\sqrt{1 + tan^{2} x}}))