Difference between revisions of "The Existence of the Exponential Function"
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+ | {{Paperlets Navigation}} | ||
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+ | ==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 | ||
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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== |
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 (with coefficients in ) which satisfies the non-linear equation
[Main] |
as well as the initial condition
[Init] |
Alternative proofs of the existence of are of course available, including the explicit formula . 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 .
The Scheme
We aim to construct and solve [Main] inductively, degree by degree. Equation [Init] gives in degrees 0 and 1, and the given formula for indeed solves [Main] in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial 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 is the "mistake for ", a certain homogeneous polynomial of degree 8 in the variables and .
Our hope is to "fix" the mistake by replacing with , where is a degree 8 "correction", a homogeneous polynomial of degree 8 in (well, in this simple case, just a multiple of ).