09/15/2013, 07:33 PM

Well I've been working a lot on continuum sums/indefinite sums and I've found a way to perhaps recover a chain rule, and with it the concept of a contour summation.

It is a linear substitution law for indefinite summation. I think I can recover a full chain rule by using luinear approximations but I am not quite sure. The rule is very simple to state and I can give a concrete reason for its result. Being quick I'll state it:

I:

A good argument for this is a few examples in integers, lets take the identity function and show it works.

But this is equal to two sums across n and m:

which follows from the division algorithm.

This substitution holds for complex numbers because the right hand side of I behaves like the left hand side when adding or

This may not seem like a big rule but it allows us to define a chain rule for linear substitutions, and what more gives us a glimmer of hope at finding a regular chain rule so that we may find a form of substitution for continuum sums.

A neat thing I've been investigating alongside this is contour summation. Imagining C is some contour in complex plane parameterized by then the contour sum is defined as:

Given if f is analytic and has an antidifference on C, then the result depends only on the end points (like contour integration). Closed contours are zero if there is an antidifference. The result does not depend on the parameterization. And I have a belief that there are non-analytic functions where there is a type of residue according to closed contours around singularities. This requires more research however.

But using our definition of the contour sum and our linear substitution there is a rule for given for circles. I wont write it all out but it gives me the thought that we can find some neat closed form formula without limits for closed contour summation. More on this as I progress. I was just wondering if anyone sees any quick applications for the linear substitution rule

It is a linear substitution law for indefinite summation. I think I can recover a full chain rule by using luinear approximations but I am not quite sure. The rule is very simple to state and I can give a concrete reason for its result. Being quick I'll state it:

I:

A good argument for this is a few examples in integers, lets take the identity function and show it works.

But this is equal to two sums across n and m:

which follows from the division algorithm.

This substitution holds for complex numbers because the right hand side of I behaves like the left hand side when adding or

This may not seem like a big rule but it allows us to define a chain rule for linear substitutions, and what more gives us a glimmer of hope at finding a regular chain rule so that we may find a form of substitution for continuum sums.

A neat thing I've been investigating alongside this is contour summation. Imagining C is some contour in complex plane parameterized by then the contour sum is defined as:

Given if f is analytic and has an antidifference on C, then the result depends only on the end points (like contour integration). Closed contours are zero if there is an antidifference. The result does not depend on the parameterization. And I have a belief that there are non-analytic functions where there is a type of residue according to closed contours around singularities. This requires more research however.

But using our definition of the contour sum and our linear substitution there is a rule for given for circles. I wont write it all out but it gives me the thought that we can find some neat closed form formula without limits for closed contour summation. More on this as I progress. I was just wondering if anyone sees any quick applications for the linear substitution rule