# infinite product expansion

• Feb 20th 2008, 03:15 AM
rollo
infinite product expansion
Hi guys, looking for some help.

I'm a hardware electronic engineer by profession rather than a pure mathematician so some of the more denser theory is not easy for me. I'm trying to produce an analytic solution for a Schwarz christoffel mapping in an electrostatics problem I'm working on.

The equation I have is this:

$\displaystyle \left(\frac{(w-u)(w+u+NL)}{(w-1)(w+1+NL)}\right)^{1/2}$

this has to be multiplied out as an infinite product in order to make the solution periodic.

I'm not really sure how to simplify/expand the above in a way so as to make the equation in the form of an infinite product, i.e. $\displaystyle \sin(z) = 1+(z/n)$ Any help would be greatly appreciated.

sorry, the above equation was wrong, fixed it now.
• Feb 20th 2008, 03:40 AM
CaptainBlack
Quote:

Originally Posted by rollo
Hi guys, looking for some help.

I'm a hardware electronic engineer by profession rather than a pure mathematician so some of the more denser theory is not easy for me. I'm trying to produce an analytic solution for a Schwarz christoffel mapping in an electrostatics problem I'm working on.

The equation I have is this:

$\displaystyle \left(\frac{(w-1)(w-u-NL)}{(w-u)(w+1+NL)}\right)^{1/2}$

this has to be multiplied out as an infinite product in order to make the solution periodic.

I'm not really sure how to simplify/expand the above in a way so as to make the equation in the form of an infinite product, i.e. $\displaystyle \sin(z) = 1+(z/n)$ Any help would be greatly appreciated.

I think that we will need a bit more explanation of what you are trying to do

RonL
• Feb 20th 2008, 04:20 AM
rollo
essentially we have a polygon structure, a rectangle with 4 corners at A,B,C,D equal to -1, -u, u, 1. This is one part of a periodic structure consiting of the same type of rectangle, shifted by a period, L. N in this case represents the number of periods along.

SC mapping is hard to attain analytical as opposed to numerical solutions above 4 vertices. In this case, in order to produce an analytic solution for the whole structure, it is enough to produce the SC mapping for one rectangle, then multiply it out as an infinite series to get the periodicity of the structure.

The above represents an SC mapping of one particular rectangle with points at u,1, -u+NL,-1+NL. I.e. it is the next rectangle shifted on from the original.

So what I need to do is to multiply out the above equation by an infinite product.

I know from a similar example (but with no access to technique) that the solution will eventually have the form of [(cos(w/l)+a)/(cos(w/l)+b)]^1/2

I can intuitively see why the cos function comes in but I can't figure out how to do this mathematically using algebra. As I said, I know that infinite products can be represented by trigonometric identities but I'm struggling to put the original equation in the correct form.

Any help greatly appreciated.