The Existence of the Exponential Function: Difference between revisions
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
{{Paperlets Navigation}} |
|||
==Introduction== |
|||
The purpose of this [[paperlet]] is to use some homological algebra in order to prove the existence of a power series <math>e(x)</math> (with coefficients in <math>{\mathbb Q}</math>) which satisfies the non-linear equation |
The purpose of this [[paperlet]] is to use some homological algebra in order to prove the existence of a power series <math>e(x)</math> (with coefficients in <math>{\mathbb Q}</math>) which satisfies the non-linear equation |
||
| Line 5: | Line 9: | ||
as well as the initial condition |
as well as the initial condition |
||
{{Equation|Init|<math>e(x)=1+x+</math>''(higher order terms)''.}} |
|||
| ⚫ | Alternative proofs of the existence of <math>e(x)</math> are of course available, including the explicit formula <math>e(x)=\sum_{k=0}^\infty\frac{x^k}{k!}</math>. Thus the value of this paperlet is not in the result it proves but rather in the story it tells: that there is a technique to solve functional equations such as {{EqRef|Main}} using homology. There are plenty of other examples for the use of that technique, in which the equation replacing {{EqRef|Main}} isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting. |
||
Thus below we will pretend not to know the exponential function and/or its relationship with the differential equation <math>e'=e</math>. |
|||
==The Scheme== |
|||
We aim to construct <math>e(x)</math> and solve {{EqRef|Main}} inductively, degree by degree. Equation {{EqRef|Init}} gives <math>e(x)</math> in degrees 0 and 1, and the given formula for <math>e(x)</math> indeed solves {{EqRef|Main}} in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial <math>e_7(x)</math> which solves {{EqRef|Main}} up to and including degree 7, but at this stage of the construction, it may well fail to solve {{EqRef|Main}} in degree 8. Thus modulo degrees 9 and up, we have |
|||
{{Equation|M|<math>e_7(x+y)-e_7(x)e_7(y)=M_8(x,y)</math>,}} |
|||
where <math>M_8(x,y)</math> is the "mistake for <math>e_7</math>", a certain homogeneous polynomial of degree 8 in the variables <math>x</math> and <math>y</math>. |
|||
Our hope is to "fix" the mistake <math>M_8</math> by replacing <math>e_7(x)</math> with <math>e_8(x)=e_7(x)+\epsilon_8(x)</math>, where <math>\epsilon_8(x)</math> is a degree 8 "correction", a homogeneous polynomial of degree 8 in <math>x</math> (well, in this simple case, just a multiple of <math>x^8</math>). |
|||
==Computing the Homology== |
|||
| ⚫ | Alternative proofs of the existence of <math>e(x)</math> are of course available, including the explicit formula <math>e(x)=\sum_{k=0}^\infty\frac{x^k}{k!}</math>. Thus the value of this |
||
Revision as of 22:09, 14 January 2007
|
Introduction
The purpose of this paperlet is to use some homological algebra in order to prove the existence of a power series [math]\displaystyle{ e(x) }[/math] (with coefficients in [math]\displaystyle{ {\mathbb Q} }[/math]) which satisfies the non-linear equation
| [Main] |
as well as the initial condition
| [Init] |
Alternative proofs of the existence of [math]\displaystyle{ e(x) }[/math] are of course available, including the explicit formula [math]\displaystyle{ e(x)=\sum_{k=0}^\infty\frac{x^k}{k!} }[/math]. Thus the value of this paperlet is not in the result it proves but rather in the story it tells: that there is a technique to solve functional equations such as [Main] using homology. There are plenty of other examples for the use of that technique, in which the equation replacing [Main] isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting.
Thus below we will pretend not to know the exponential function and/or its relationship with the differential equation [math]\displaystyle{ e'=e }[/math].
The Scheme
We aim to construct [math]\displaystyle{ e(x) }[/math] and solve [Main] inductively, degree by degree. Equation [Init] gives [math]\displaystyle{ e(x) }[/math] in degrees 0 and 1, and the given formula for [math]\displaystyle{ e(x) }[/math] indeed solves [Main] in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial [math]\displaystyle{ e_7(x) }[/math] which solves [Main] up to and including degree 7, but at this stage of the construction, it may well fail to solve [Main] in degree 8. Thus modulo degrees 9 and up, we have
| [M] |
where [math]\displaystyle{ M_8(x,y) }[/math] is the "mistake for [math]\displaystyle{ e_7 }[/math]", a certain homogeneous polynomial of degree 8 in the variables [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math].
Our hope is to "fix" the mistake [math]\displaystyle{ M_8 }[/math] by replacing [math]\displaystyle{ e_7(x) }[/math] with [math]\displaystyle{ e_8(x)=e_7(x)+\epsilon_8(x) }[/math], where [math]\displaystyle{ \epsilon_8(x) }[/math] is a degree 8 "correction", a homogeneous polynomial of degree 8 in [math]\displaystyle{ x }[/math] (well, in this simple case, just a multiple of [math]\displaystyle{ x^8 }[/math]).
