Integration in Finite Terms

Please download to get full document.

View again

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
 3
 
 

Slides

  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…
Related documents
Share
Transcript
  • 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
  • Related Search
    We Need Your Support
    Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

    Thanks to everyone for your continued support.

    No, Thanks