Difference between revisions of "The Existence of the Exponential Function"

From Drorbn
Jump to: navigation, search
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
  
<center><math>e(x)=1+x+</math>''(higher order terms)''.</center>
+
{{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>).
  
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.
+
==Computing the Homology==

Revision as of 23:09, 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 e(x) (with coefficients in {\mathbb Q}) which satisfies the non-linear equation

[Main]
e(x+y)=e(x)e(y)

as well as the initial condition

[Init]
e(x)=1+x+(higher order terms).

Alternative proofs of the existence of e(x) are of course available, including the explicit formula e(x)=\sum_{k=0}^\infty\frac{x^k}{k!}. 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 e'=e.

The Scheme

We aim to construct e(x) and solve [Main] inductively, degree by degree. Equation [Init] gives e(x) in degrees 0 and 1, and the given formula for e(x) indeed solves [Main] in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial e_7(x) 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]
e_7(x+y)-e_7(x)e_7(y)=M_8(x,y),

where M_8(x,y) is the "mistake for e_7", a certain homogeneous polynomial of degree 8 in the variables x and y.

Our hope is to "fix" the mistake M_8 by replacing e_7(x) with e_8(x)=e_7(x)+\epsilon_8(x), where \epsilon_8(x) is a degree 8 "correction", a homogeneous polynomial of degree 8 in x (well, in this simple case, just a multiple of x^8).

Computing the Homology