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Integration in finite terms Formalizing the question Applications Sources Technicalities e x 2 dx ? Why cant we do Non impeditus ab ulla scientia K. P. Hart Faculty EEMCS TU Delft Delft, 10 November, 2006: 16:0017:00 e x

  1. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx ? � Why can’t we do Non impeditus ab ulla scientia K. P. Hart Faculty EEMCS TU Delft Delft, 10 November, 2006: 16:00–17:00 e − x 2 dx ? K. P. Hart Why can’t we do R

  2. Integration in finite terms Formalizing the question Applications Sources Technicalities Outline Integration in finite terms 1 Formalizing the question 2 Differential fields Elementary extensions The abstract formulation Applications 3 Liouville’s criterion e − z 2 dz at last � Further examples Sources 4 Technicalities 5 e − x 2 dx ? K. P. Hart Why can’t we do R

  3. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx ’ mean? � What does ‘do � To ‘do’ an (indefinite) integral f ( x ) dx , means to find a formula, F ( x ), however nasty, such that F ′ = f . e − x 2 dx ? K. P. Hart Why can’t we do R

  4. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx ’ mean? � What does ‘do � 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? e − x 2 dx ? K. P. Hart Why can’t we do R

  5. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx ’ mean? � What does ‘do � 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? e − x 2 dx ? K. P. Hart Why can’t we do R

  6. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx ’ mean? � What does ‘do � 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? e − x 2 dx cannot be done? � How do we then prove that e − x 2 dx ? K. P. Hart Why can’t we do R

  7. Integration in finite terms Formalizing the question Applications Sources Technicalities What is a formula? We recognise a formula when we see one. e − x 2 dx ? K. P. Hart Why can’t we do R

  8. Integration in finite terms Formalizing the question Applications Sources Technicalities What is a formula? We recognise a formula when we see one. e − x 2 dx does not count, because � E.g., Maple’s answer to e − x 2 dx ? K. P. Hart Why can’t we do R

  9. Integration in finite terms Formalizing the question Applications Sources Technicalities What is a formula? We recognise a formula when we see one. e − x 2 dx does not count, because � E.g., Maple’s answer to 1 √ π erf( x ) 2 e − x 2 dx ? K. P. Hart Why can’t we do R

  10. Integration in finite terms Formalizing the question Applications Sources Technicalities What is a formula? We recognise a formula when we see one. e − x 2 dx does not count, because � E.g., Maple’s answer to 1 √ π erf( x ) 2 is simply an abbreviation for ‘a primitive function of e − x 2 ’ e − x 2 dx ? K. P. Hart Why can’t we do R

  11. Integration in finite terms Formalizing the question Applications Sources Technicalities What is a formula? We recognise a formula when we see one. e − x 2 dx does not count, because � E.g., Maple’s answer to 1 √ π erf( x ) 2 is simply an abbreviation for ‘a primitive function of e − x 2 ’ (see Maple’s help facility). e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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, . . . e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 , . . . e − x 2 dx ? K. P. Hart Why can’t we do R

  17. Integration in finite terms Formalizing the question Applications Sources Technicalities Can we formalize that? Yes. e − x 2 dx ? K. P. Hart Why can’t we do R

  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, e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  22. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx cannot be done? � How do we then prove that We do not look at all functions that we get in this way and check that their derivatives are not e − x 2 . e − x 2 dx ? K. P. Hart Why can’t we do R

  23. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx cannot be done? � How do we then prove that We do not look at all functions that we get in this way and check that their derivatives are not e − x 2 . 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. e − x 2 dx ? K. P. Hart Why can’t we do R

  24. Integration in finite terms Formalizing the question Applications Sources Technicalities e − x 2 dx cannot be done? � How do we then prove that We do not look at all functions that we get in this way and check that their derivatives are not e − x 2 . 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 − x 2 does not satisfy this condition. e − x 2 dx ? K. P. Hart Why can’t we do R

  25. Integration in finite terms Formalizing the question Differential fields Applications Elementary extensions Sources The abstract formulation Technicalities Outline Integration in finite terms 1 Formalizing the question 2 Differential fields Elementary extensions The abstract formulation Applications 3 Liouville’s criterion e − z 2 dz at last � Further examples Sources 4 Technicalities 5 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 e − x 2 dx ? K. P. Hart Why can’t we do R

  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 ) e − x 2 dx ? K. P. Hart Why can’t we do R

  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 ) e − x 2 dx ? K. P. Hart Why can’t we do R

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