[PPT] - e x 2 dx ? Why cant we do Non impeditus ab ulla scientia K. P. PowerPoint Presentation

SLIDE 1

Integration in finite terms Formalizing the question Applications Sources Technicalities

Why can’t we do

e−x2 dx?

Non impeditus ab ulla scientia

K. P. Hart

Faculty EEMCS TU Delft

Delft, 10 November, 2006: 16:00–17:00

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 2

Integration in finite terms Formalizing the question Applications Sources Technicalities

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 3

Integration in finite terms Formalizing the question Applications Sources Technicalities

What does ‘do

e−x2 dx’ mean?

To ‘do’ an (indefinite) integral

f (x) dx, means to find a formula,

F(x), however nasty, such that F ′ = f .

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 4

Integration in finite terms Formalizing the question Applications Sources Technicalities

What does ‘do

e−x2 dx’ mean?

To ‘do’ an (indefinite) integral

f (x) dx, means to find a formula,

F(x), however nasty, such that F ′ = f . What is a formula?

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 5

Integration in finite terms Formalizing the question Applications Sources Technicalities

What does ‘do

e−x2 dx’ mean?

To ‘do’ an (indefinite) integral

f (x) dx, means to find a formula,

F(x), however nasty, such that F ′ = f . What is a formula? Can we formalize that?

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 6

Integration in finite terms Formalizing the question Applications Sources Technicalities

What does ‘do

e−x2 dx’ mean?

To ‘do’ an (indefinite) integral

f (x) dx, means to find a formula,

F(x), however nasty, such that F ′ = f . What is a formula? Can we formalize that? How do we then prove that

e−x2 dx cannot be done?
K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 7

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

We recognise a formula when we see one.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 8

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

We recognise a formula when we see one. E.g., Maple’s answer to

e−x2 dx does not count, because
K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 9

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

We recognise a formula when we see one. E.g., Maple’s answer to

e−x2 dx does not count, because

1 2 √π erf(x)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 10

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

We recognise a formula when we see one. E.g., Maple’s answer to

e−x2 dx does not count, because

1 2 √π erf(x) is simply an abbreviation for ‘a primitive function of e−x2’

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 11

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

We recognise a formula when we see one. E.g., Maple’s answer to

e−x2 dx does not count, because

1 2 √π erf(x) is simply an abbreviation for ‘a primitive function of e−x2’ (see Maple’s help facility).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 12

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

A formula is an expression built up from elementary functions using only

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 13

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

A formula is an expression built up from elementary functions using only addition, multiplication, . . .

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 14

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

A formula is an expression built up from elementary functions using only addition, multiplication, . . .

ther algebra: roots ’n such
K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 15

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

A formula is an expression built up from elementary functions using only addition, multiplication, . . .

ther algebra: roots ’n such

composition of functions

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 16

Integration in finite terms Formalizing the question Applications Sources Technicalities

What is a formula?

A formula is an expression built up from elementary functions using only addition, multiplication, . . .

ther algebra: roots ’n such

composition of functions Elementary functions: ex, sin x, x, log x, . . .

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 17

Integration in finite terms Formalizing the question Applications Sources Technicalities

Can we formalize that?

Yes.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 18

Integration in finite terms Formalizing the question Applications Sources Technicalities

Can we formalize that?

Yes. Start with C(z) the field of (complex) rational functions and add, one at a time,

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 19

Integration in finite terms Formalizing the question Applications Sources Technicalities

Can we formalize that?

Yes. Start with C(z) the field of (complex) rational functions and add, one at a time, algebraic elements

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 20

Integration in finite terms Formalizing the question Applications Sources Technicalities

Can we formalize that?

Yes. Start with C(z) the field of (complex) rational functions and add, one at a time, algebraic elements logarithms

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 21

Integration in finite terms Formalizing the question Applications Sources Technicalities

Can we formalize that?

Yes. Start with C(z) the field of (complex) rational functions and add, one at a time, algebraic elements logarithms exponentials

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 22

Integration in finite terms Formalizing the question Applications Sources Technicalities

How do we then prove that

e−x2 dx cannot be done?

We do not look at all functions that we get in this way and check that their derivatives are not e−x2.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 23

Integration in finite terms Formalizing the question Applications Sources Technicalities

How do we then prove that

e−x2 dx cannot be done?

We do not look at all functions that we get in this way and check that their derivatives are not e−x2. We do establish an algebraic condition for a function to have a primitive function that is expressible in terms of elementary functions, as described above.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 24

Integration in finite terms Formalizing the question Applications Sources Technicalities

How do we then prove that

e−x2 dx cannot be done?

We do not look at all functions that we get in this way and check that their derivatives are not e−x2. We do establish an algebraic condition for a function to have a primitive function that is expressible in terms of elementary functions, as described above. We then show that e−x2 does not satisfy this condition.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 25

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 26

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A differential field is a field F with a derivation, that is, a map D : F → F that satisfies

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 27

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A differential field is a field F with a derivation, that is, a map D : F → F that satisfies D(a + b) = D(a) + D(b)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 28

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A differential field is a field F with a derivation, that is, a map D : F → F that satisfies D(a + b) = D(a) + D(b) D(ab) = D(a)b + aD(b)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 29

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Main example(s)

The rational (meromorphic) functions on (some domain in) C, with D(f ) = f ′ (of course).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 30

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Main example(s)

The rational (meromorphic) functions on (some domain in) C, with D(f ) = f ′ (of course). We write a′ = D(a) in any differential field.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 31

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Easy properties

Exercises (an)′ = nan−1a′

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 32

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Easy properties

Exercises (an)′ = nan−1a′ (a/b)′ = (a′b −ab′)/b2 (Hint: f = a/b solve (bf )′ = a′ for f ′)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 33

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Easy properties

Exercises (an)′ = nan−1a′ (a/b)′ = (a′b −ab′)/b2 (Hint: f = a/b solve (bf )′ = a′ for f ′) 1′ = 0 (Hint: 1′ = (12)′)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 34

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Easy properties

Exercises (an)′ = nan−1a′ (a/b)′ = (a′b −ab′)/b2 (Hint: f = a/b solve (bf )′ = a′ for f ′) 1′ = 0 (Hint: 1′ = (12)′) The ‘constants’, i.e., the c ∈ F with c′ = 0 form a subfield

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 35

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Exponentials and logarithms

a is an exponential of b if a′ = b′a

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 36

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Exponentials and logarithms

a is an exponential of b if a′ = b′a b is a logarithm of a if b′ = a′/a

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 37

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Exponentials and logarithms

a is an exponential of b if a′ = b′a b is a logarithm of a if b′ = a′/a so: a is an exponential of b iff b is a logarithm of a.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 38

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Exponentials and logarithms

a is an exponential of b if a′ = b′a b is a logarithm of a if b′ = a′/a so: a is an exponential of b iff b is a logarithm of a. ‘logarithmic derivative’: (ambn)′ ambn = ma′ a + nb′ b

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 39

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Exponentials and logarithms

a is an exponential of b if a′ = b′a b is a logarithm of a if b′ = a′/a so: a is an exponential of b iff b is a logarithm of a. ‘logarithmic derivative’: (ambn)′ ambn = ma′ a + nb′ b Much of Calculus is actually Algebra . . .

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 40

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 41

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A simple elementary extension of a differential field F is a field extension F(t) where t is algebraic over F,

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 42

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A simple elementary extension of a differential field F is a field extension F(t) where t is algebraic over F, an exponential of some b ∈ F, or

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 43

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A simple elementary extension of a differential field F is a field extension F(t) where t is algebraic over F, an exponential of some b ∈ F, or a logarithm of some a ∈ F

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 44

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Definition

A simple elementary extension of a differential field F is a field extension F(t) where t is algebraic over F, an exponential of some b ∈ F, or a logarithm of some a ∈ F G is an elementary extension of F is G = F(t1, t2, . . . , tN), where each time Fi(ti+1) is a simple elementary extension of Fi = F(t1, . . . , ti).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 45

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 46

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Elementary integrals

We say that a ∈ F has an elementary integral if there is an elementary extension G of F with an element t such that t′ = a.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 47

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Elementary integrals

We say that a ∈ F has an elementary integral if there is an elementary extension G of F with an element t such that t′ = a. The Question: characterize (of give necessary conditions for) this.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 48

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

A characterization

Theorem (Rosenlicht) Let F be a differential field of characteristic zero and a ∈ F. If a has an elementary integral in some extension with the same field of constants then there are v ∈ F, constants c1, . . . , cn ∈ F and elements u1, . . . un ∈ F such that a = v′ + c1 u′

1

u1 + · · · + cn u′

n

un .

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 49

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

A characterization

Theorem (Rosenlicht) Let F be a differential field of characteristic zero and a ∈ F. If a has an elementary integral in some extension with the same field of constants then there are v ∈ F, constants c1, . . . , cn ∈ F and elements u1, . . . un ∈ F such that a = v′ + c1 u′

1

u1 + · · · + cn u′

n

un . The converse is also true: a = (v + c1 log u1 + · · · + cn log un)′.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 50

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Comment on the constants

Consider

1 1+x2 ∈ R(x)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 51

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Comment on the constants

Consider

1 1+x2 ∈ R(x)

an elementary integral is 1 2i ln x − i x + i

,

using a larger field of constants: C

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 52

Integration in finite terms Formalizing the question Applications Sources Technicalities Differential fields Elementary extensions The abstract formulation

Comment on the constants

Consider

1 1+x2 ∈ R(x)

an elementary integral is 1 2i ln x − i x + i

,

using a larger field of constants: C there are no v, ui and ci in R(x) as in Rosenlicht’s theorem.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 53

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 54

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

When can we do

f (z)eg(z) dz?

Let f and g be rational functions over C, with f nonzero and g non-constant.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 55

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

When can we do

f (z)eg(z) dz?

Let f and g be rational functions over C, with f nonzero and g non-constant. feg belongs to the field F = C(z, t), where t = eg (and t′ = gt).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 56

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

When can we do

f (z)eg(z) dz?

Let f and g be rational functions over C, with f nonzero and g non-constant. feg belongs to the field F = C(z, t), where t = eg (and t′ = gt). F is a transcendental extension of C(z).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 57

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

When can we do

f (z)eg(z) dz?

Let f and g be rational functions over C, with f nonzero and g non-constant. feg belongs to the field F = C(z, t), where t = eg (and t′ = gt). F is a transcendental extension of C(z). If feg has an elementary integral then in F we must have ft = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ C(z, t).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 58

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

The criterion

Using algebraic considerations one can then get the following criterion. Theorem (Liouville) The function feg has an elementary integral iff there is a rational function q ∈ C(z) such that f = q′ + qg′ the integral then is qeg (of course).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 59

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 60

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

e−z2 dz

In this case f (z) = 1 and g(z) = −z2.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 61

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

e−z2 dz

In this case f (z) = 1 and g(z) = −z2. Is there a q such that 1 = q′(z) − 2zq(z)?

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 62

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

e−z2 dz

In this case f (z) = 1 and g(z) = −z2. Is there a q such that 1 = q′(z) − 2zq(z)? Assume q has a pole β and look at principal part of Laurent series

m

i=1

αi (z − β)i Its contribution to the right-hand side should be zero.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 63

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

e−z2 dz

We get, at the pole β: 0 =

m

i=1
−

iαi (z − β)i+1 − 2zαi (z − β)i

Successively: α1 = 0, . . . , αm = 0.
K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 64

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

e−z2 dz

We get, at the pole β: 0 =

m

i=1
−

iαi (z − β)i+1 − 2zαi (z − β)i

Successively: α1 = 0, . . . , αm = 0.

So, q is a polynomial, but 1 = q′(z) − 2zq(z) will give a mismatch

f degrees.
K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 65

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

Outline

1

Integration in finite terms

2

Formalizing the question Differential fields Elementary extensions The abstract formulation

3

Applications Liouville’s criterion

e−z2 dz at last

Further examples

4

Sources

5

Technicalities

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 66

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

ez

z dz

Here f (z) = 1/z and g(z) = z, so we need q(z) such that 1 z = q′(z) + q(z) Again, via partial fractions: no such q exists.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 67

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

ez

z dz

Here f (z) = 1/z and g(z) = z, so we need q(z) such that 1 z = q′(z) + q(z) Again, via partial fractions: no such q exists.

eez dz =

eu

u du =

1

ln v dv

(substitutions: u = ez and u = ln v)

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 68

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

sin z

z dz

In the complex case this is just ez−e−z

z

dz.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 69

Integration in finite terms Formalizing the question Applications Sources Technicalities Liouville’s criterion R e−z2 dz at last Further examples

sin z

z dz

In the complex case this is just ez−e−z

z

dz. Let t = ez and work in C(z, t); adapt the proof of the main theorem to reduce this to 1

z = q′(z) + q(z) with q ∈ C(z), still

impossible.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 70

Integration in finite terms Formalizing the question Applications Sources Technicalities

Light reading

These slides at: fa.its.tudelft.nl/~hart

J. Liouville.

M´ emoire sur les transcendents elliptiques consid´ er´ ees comme functions de leur amplitudes, Journal d’´ Ecole Royale Polytechnique (1834)

M. Rosenlicht,

Integration in finite terms, American Mathematical Monthly, 79 (1972), 963–972.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 71

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, I

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′ ∈ F.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 72

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, I

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′ ∈ F. Let f (t) ∈ F[t] be a polynomial of positive degree.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 73

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, I

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′ ∈ F. Let f (t) ∈ F[t] be a polynomial of positive degree. Then f (t)′ is a polynomial in F[t] of the same degree as f (t) or

ne less, depending on whether the leading coefficient of f (t) is

not, or is, a constant.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 74

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, II

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′/t ∈ F.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 75

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, II

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′/t ∈ F. Let f (t) ∈ F[t] be a polynomial of positive degree.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 76

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, II

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′/t ∈ F. Let f (t) ∈ F[t] be a polynomial of positive degree. for nonzero a ∈ F and nonzero n ∈ Z we have (atn)′ = htn for some nonzero h ∈ F;

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 77

Integration in finite terms Formalizing the question Applications Sources Technicalities

A useful lemma, II

Lemma Let F be a differential field, F(t) a differential extension with the same constants, with t transcendental over F and such that t′/t ∈ F. Let f (t) ∈ F[t] be a polynomial of positive degree. for nonzero a ∈ F and nonzero n ∈ Z we have (atn)′ = htn for some nonzero h ∈ F; if f (t) ∈ F[t] then f (t)′ is of the same degree as f (t) and f (t)′ is a multiple of f (t) iff f (t) is a monomial (atn).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 78

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, I

Write F = C(z) and t = ez.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 79

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, I

Write F = C(z) and t = ez. If sin z

z

dz were elementary then t2 − 1 tz = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ F(t).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 80

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, I

Write F = C(z) and t = ez. If sin z

z

dz were elementary then t2 − 1 tz = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ F(t). By logarithmic differentiation: the ui’s not in F are monic and irreducible in F[t].

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 81

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, II

If sin z

z

dz were elementary then t2 − 1 tz = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ F(t).

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 82

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, II

If sin z

z

dz were elementary then t2 − 1 tz = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ F(t). By the lemma just one ui is not in F and this ui is t.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 83

Integration in finite terms Formalizing the question Applications Sources Technicalities

sin x

x dz, II

If sin z

z

dz were elementary then t2 − 1 tz = v′ + c1 u′

1

u1 + · · · + cn u′

n

un with c1, . . . , cn ∈ C and v, u1, . . . , un ∈ F(t). By the lemma just one ui is not in F and this ui is t. So c1

u′

1

u1 + · · · + cn u′

n

un is in F.

K. P. Hart

Why can’t we do R e−x2 dx?

SLIDE 84

Integration in finite terms Formalizing the question Applications Sources Technicalities