1.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
Why can’t we do e −x dx?
Non impeditus ab ulla scientia
K. P. Hart
Faculty EEMCS
TU Delft
Delft, 10 November, 2006: 16:00–17:00
R −x 2
K. P. Hart Why can’t we do e dx?
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
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
3.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
What does ‘do e −x dx’ mean?
To ‘do’ an (indeﬁnite) integral f (x) dx, means to ﬁnd a formula,
F (x), however nasty, such that F = f .
R −x 2
K. P. Hart Why can’t we do e dx?
4.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
What does ‘do e −x 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?
R −x 2
K. P. Hart Why can’t we do e dx?
5.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
What does ‘do e −x 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?
R −x 2
K. P. Hart Why can’t we do e dx?
6.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
What does ‘do e −x 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?
2
How do we then prove that e −x dx cannot be done?
R −x 2
K. P. Hart Why can’t we do e dx?
7.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
What is a formula?
We recognise a formula when we see one.
R −x 2
K. P. Hart Why can’t we do e dx?
8.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
What is a formula?
We recognise a formula when we see one.
2
E.g., Maple’s answer to e −x dx does not count, because
R −x 2
K. P. Hart Why can’t we do e dx?
9.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
What is a formula?
We recognise a formula when we see one.
2
E.g., Maple’s answer to e −x dx does not count, because
1√
π erf(x)
2
R −x 2
K. P. Hart Why can’t we do e dx?
10.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
What is a formula?
We recognise a formula when we see one.
2
E.g., Maple’s answer to e −x dx does not count, because
1√
π erf(x)
2
2
is simply an abbreviation for ‘a primitive function of e −x ’
R −x 2
K. P. Hart Why can’t we do e dx?
11.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
What is a formula?
We recognise a formula when we see one.
2
E.g., Maple’s answer to e −x dx does not count, because
1√
π erf(x)
2
2
is simply an abbreviation for ‘a primitive function of e −x ’
(see Maple’s help facility).
R −x 2
K. P. Hart Why can’t we do e dx?
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
R −x 2
K. P. Hart Why can’t we do e dx?
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, . . .
R −x 2
K. P. Hart Why can’t we do e dx?
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, . . .
other algebra: roots ’n such
R −x 2
K. P. Hart Why can’t we do e dx?
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, . . .
other algebra: roots ’n such
composition of functions
R −x 2
K. P. Hart Why can’t we do e dx?
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, . . .
other algebra: roots ’n such
composition of functions
Elementary functions: e x , sin x, x, log x, . . .
R −x 2
K. P. Hart Why can’t we do e dx?
17.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
Can we formalize that?
Yes.
R −x 2
K. P. Hart Why can’t we do e dx?
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,
R −x 2
K. P. Hart Why can’t we do e dx?
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
R −x 2
K. P. Hart Why can’t we do e dx?
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
R −x 2
K. P. Hart Why can’t we do e dx?
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
R −x 2
K. P. Hart Why can’t we do e dx?
22.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
How do we then prove that e −x dx cannot be done?
We do not look at all functions that we get in this way and check
2
that their derivatives are not e −x .
R −x 2
K. P. Hart Why can’t we do e dx?
23.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
How do we then prove that e −x dx cannot be done?
We do not look at all functions that we get in this way and check
2
that their derivatives are not e −x .
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.
R −x 2
K. P. Hart Why can’t we do e dx?
24.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
2
How do we then prove that e −x dx cannot be done?
We do not look at all functions that we get in this way and check
2
that their derivatives are not e −x .
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.
2
We then show that e −x does not satisfy this condition.
R −x 2
K. P. Hart Why can’t we do e dx?
25.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
26.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Deﬁnition
A diﬀerential ﬁeld is a ﬁeld F with a derivation, that is, a map
D : F → F that satisﬁes
R −x 2
K. P. Hart Why can’t we do e dx?
27.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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)
R −x 2
K. P. Hart Why can’t we do e dx?
28.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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)
R −x 2
K. P. Hart Why can’t we do e dx?
29.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Main example(s)
The rational (meromorphic) functions on (some domain in) C,
with D(f ) = f (of course).
R −x 2
K. P. Hart Why can’t we do e dx?
30.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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.
R −x 2
K. P. Hart Why can’t we do e dx?
31.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Easy properties
Exercises
(an ) = nan−1 a
R −x 2
K. P. Hart Why can’t we do e dx?
32.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Easy properties
Exercises
(an ) = nan−1 a
(a/b) = (ab − a b)/b 2 (Hint: f = a/b solve (bf ) = a for f )
R −x 2
K. P. Hart Why can’t we do e dx?
33.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Easy properties
Exercises
(an ) = nan−1 a
(a/b) = (ab − a b)/b 2 (Hint: f = a/b solve (bf ) = a for f )
1 = 0 (Hint: 1 = (12 ) )
R −x 2
K. P. Hart Why can’t we do e dx?
34.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Easy properties
Exercises
(an ) = nan−1 a
(a/b) = (ab − a b)/b 2 (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
R −x 2
K. P. Hart Why can’t we do e dx?
35.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Exponentials and logarithms
a is an exponential of b if a = b a
R −x 2
K. P. Hart Why can’t we do e dx?
36.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Exponentials and logarithms
a is an exponential of b if a = b a
b is a logarithm of a if b = a /a
R −x 2
K. P. Hart Why can’t we do e dx?
37.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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.
R −x 2
K. P. Hart Why can’t we do e dx?
38.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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’:
(am b n ) a b
m bn
=m +n
a a b
R −x 2
K. P. Hart Why can’t we do e dx?
39.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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’:
(am b n ) a b
m bn
=m +n
a a b
Much of Calculus is actually Algebra . . .
R −x 2
K. P. Hart Why can’t we do e dx?
40.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
41.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Deﬁnition
A simple elementary extension of a diﬀerential ﬁeld F is a ﬁeld
extension F (t) where t is
algebraic over F ,
R −x 2
K. P. Hart Why can’t we do e dx?
42.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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
R −x 2
K. P. Hart Why can’t we do e dx?
43.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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
R −x 2
K. P. Hart Why can’t we do e dx?
44.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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 ).
R −x 2
K. P. Hart Why can’t we do e dx?
45.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
46.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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.
R −x 2
K. P. Hart Why can’t we do e dx?
47.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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.
R −x 2
K. P. Hart Why can’t we do e dx?
48.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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
u1 u
a = v + c1 + · · · + cn n .
u1 un
R −x 2
K. P. Hart Why can’t we do e dx?
49.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
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
u1 u
a = v + c1 + · · · + cn n .
u1 un
The converse is also true: a = (v + c1 log u1 + · · · + cn log un ) .
R −x 2
K. P. Hart Why can’t we do e dx?
50.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Comment on the constants
1
Consider 1+x 2
∈ R(x)
R −x 2
K. P. Hart Why can’t we do e dx?
51.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Comment on the constants
1
Consider 1+x 2
∈ R(x)
an elementary integral is
1 x −i
ln ,
2i x +i
using a larger ﬁeld of constants: C
R −x 2
K. P. Hart Why can’t we do e dx?
52.
Integration in ﬁnite terms
Formalizing the question Diﬀerential ﬁelds
Applications Elementary extensions
Sources The abstract formulation
Technicalities
Comment on the constants
1
Consider 1+x 2
∈ R(x)
an elementary integral is
1 x −i
ln ,
2i x +i
using a larger ﬁeld of constants: C
there are no v , ui and ci in R(x) as in Rosenlicht’s theorem.
R −x 2
K. P. Hart Why can’t we do e dx?
53.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
54.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
When can we do f (x)e g (x) , dx?
Let f and g be rational functions over C, with f nonzero and g
non-constant.
R −x 2
K. P. Hart Why can’t we do e dx?
55.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
When can we do f (x)e g (x) , dx?
Let f and g be rational functions over C, with f nonzero and g
non-constant.
fe g belongs to the ﬁeld F = C (z, t), where t = e g (and t = gt).
R −x 2
K. P. Hart Why can’t we do e dx?
56.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
When can we do f (x)e g (x) , dx?
Let f and g be rational functions over C, with f nonzero and g
non-constant.
fe g belongs to the ﬁeld F = C (z, t), where t = e g (and t = gt).
F is a transcendental extension of C(z).
R −x 2
K. P. Hart Why can’t we do e dx?
57.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
When can we do f (x)e g (x) , dx?
Let f and g be rational functions over C, with f nonzero and g
non-constant.
fe g belongs to the ﬁeld F = C (z, t), where t = e g (and t = gt).
F is a transcendental extension of C(z).
If fe g has an elementary integral then in F we must have
u1 u
ft = v + c1 + · · · + cn n
u1 un
with c1 , . . . , cn ∈ C and v , u1 , . . . , un ∈ C(z, t).
R −x 2
K. P. Hart Why can’t we do e dx?
58.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
The criterion
Using algebraic considerations one can then get the following
criterion.
Theorem (Liouville)
The function fe g has an elementary integral iﬀ there is a rational
function q ∈ C(z) such that
f = q + qg
the integral then is qe g (of course).
R −x 2
K. P. Hart Why can’t we do e dx?
59.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
60.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
2
e −z dz
In this case f (z) = 1 and g (z) = −z 2 .
R −x 2
K. P. Hart Why can’t we do e dx?
61.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
2
e −z dz
In this case f (z) = 1 and g (z) = −z 2 .
Is there a q such that 1 = q (z) − 2zq(z)?
R −x 2
K. P. Hart Why can’t we do e dx?
62.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
2
e −z dz
In this case f (z) = 1 and g (z) = −z 2 .
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
αi
(z − β)i
i=1
Its contribution to the right-hand side should be zero.
R −x 2
K. P. Hart Why can’t we do e dx?
63.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
2
e −z dz
We get, at the pole β:
m
iαi 2zαi
0= − i+1
−
(z − β) (z − β)i
i=1
Successively: α1 = 0, . . . , αm = 0.
R −x 2
K. P. Hart Why can’t we do e dx?
64.
Integration in ﬁnite terms
Formalizing the question Liouville’s criterion
R −z 2
Applications e dz at last
Sources Further examples
Technicalities
2
e −z dz
We get, at the pole β:
m
iαi 2zαi
0= − i+1
−
(z − β) (z − β)i
i=1
Successively: α1 = 0, . . . , αm = 0.
So, q is a polynomial, but 1 = q (z) − 2zq(z) will give a mismatch
of degrees.
R −x 2
K. P. Hart Why can’t we do e dx?
65.
Integration in ﬁnite terms
Formalizing the question R −z 2 criterion
Liouville’s
Applications e dz at last
Sources Further examples
Technicalities
Outline
1 Integration in ﬁnite terms
2 Formalizing the question
Diﬀerential ﬁelds
Elementary extensions
The abstract formulation
3 Applications
Liouville’s criterion
2
e −z dz at last
Further examples
4 Sources
5 Technicalities
R −x 2
K. P. Hart Why can’t we do e dx?
66.
Integration in ﬁnite terms
Formalizing the question R −z 2 criterion
Liouville’s
Applications e dz at last
Sources Further examples
Technicalities
ez
z dx
Here f (z) = 1/z and g (z) = z, so we need q(z) such that
1
= q (z) + q(z)
z
Again, via partial fractions: no such q exists.
R −x 2
K. P. Hart Why can’t we do e dx?
67.
Integration in ﬁnite terms
Formalizing the question R −z 2 criterion
Liouville’s
Applications e dz at last
Sources Further examples
Technicalities
ez
z dx
Here f (z) = 1/z and g (z) = z, so we need q(z) such that
1
= q (z) + q(z)
z
Again, via partial fractions: no such q exists.
z u
e e dz = eu du = ln v dv1
(substitutions: u = e z and u = ln v )
R −x 2
K. P. Hart Why can’t we do e dx?
68.
Integration in ﬁnite terms
Formalizing the question R −z 2 criterion
Liouville’s
Applications e dz at last
Sources Further examples
Technicalities
sin z
z dx
e z −e −z
In the complex case this is just z dz.
R −x 2
K. P. Hart Why can’t we do e dx?
69.
Integration in ﬁnite terms
Formalizing the question R −z 2 criterion
Liouville’s
Applications e dz at last
Sources Further examples
Technicalities
sin z
z dx
z −z
In the complex case this is just e −e dz.
z
Let t = e z and work in C(z, t); adapt the proof of the main
1
theorem to reduce this to z = q (z) + q(z) with q ∈ C(z), still
impossible.
R −x 2
K. P. Hart Why can’t we do e dx?
70.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
Light reading
These slides at: fa.its.tudelft.nl/~hart
J. Liouville.
M´moire sur les transcendents elliptiques consid´r´es comme
e ee
´
functions de leur amplitudes, Journal d’Ecole Royale
Polytechnique (1834)
M. Rosenlicht,
Integration in ﬁnite terms, American Mathematical Monthly,
79 (1972), 963–972.
R −x 2
K. P. Hart Why can’t we do e dx?
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 .
R −x 2
K. P. Hart Why can’t we do e dx?
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.
R −x 2
K. P. Hart Why can’t we do e dx?
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
one less, depending on whether the leading coeﬃcient of f (t) is
not, or is, a constant.
R −x 2
K. P. Hart Why can’t we do e dx?
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 .
R −x 2
K. P. Hart Why can’t we do e dx?
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.
R −x 2
K. P. Hart Why can’t we do e dx?
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 (at n ) = ht n
for some nonzero h ∈ F ;
R −x 2
K. P. Hart Why can’t we do e dx?
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 (at n ) = ht n
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 (at n ).
R −x 2
K. P. Hart Why can’t we do e dx?
78.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, I
Write F = C(z) and t = e z .
R −x 2
K. P. Hart Why can’t we do e dx?
79.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, I
Write F = C(z) and t = e z .
If sin z dz were elementary then
z
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 un
with c1 , . . . , cn ∈ C and v , u1 , . . . , un ∈ F (t).
R −x 2
K. P. Hart Why can’t we do e dx?
80.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, I
Write F = C(z) and t = e z .
If sin z dz were elementary then
z
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 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].
R −x 2
K. P. Hart Why can’t we do e dx?
81.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, II
sin z
If z dz were elementary then
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 un
with c1 , . . . , cn ∈ C and v , u1 , . . . , un ∈ F (t).
R −x 2
K. P. Hart Why can’t we do e dx?
82.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, II
sin z
If z dz were elementary then
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 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.
R −x 2
K. P. Hart Why can’t we do e dx?
83.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, II
sin z
If z dz were elementary then
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 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.
u1
So c1 u1 + · · · + cn un is in F .
un
R −x 2
K. P. Hart Why can’t we do e dx?
84.
Integration in ﬁnite terms
Formalizing the question
Applications
Sources
Technicalities
sin x
x dz, III
Finally, in
t2 − 1 u u
= v + c1 1 + · · · + cn n
tz u1 un
1
we must have v = bj t j and from this: z = b1 + b1 .
R −x 2
K. P. Hart Why can’t we do e dx?
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