Integration in Finite Terms

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Why some elementary functions have no elementary integral

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Integration in Finite Terms

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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