How to mathematically express sum of products

[szg07]

Hi everybody,

I have been solving some problems in physics and mathematics that often involve expressions where you need to represent the sum of products,

a simple example would be :

resistors in parallel

when 3 resistors are in parallel, you always get an expression like :

(R1*R2*R3)/ (R1*R2 + R1*R3 + R2*R3)

and when you have an n resistors in parallel, that would give :

(R1*R2*R3*...*Rn) / (the sum of all combinations of (n-1) resistors out of n resistors)

my problem is that I would like to be able to represent such results with a general formula
using the SIGMA sign or PI sign whatever the number of resistors involved

and even with matrices (determinant)

How would I do that?

Hope you can help me!

Thanks a lot

 

[Hootenanny]

Welcome to Physics Forums.

In your resistor example, one could write:

$$\sum_{\begin{array}{c}j,k=1\\ j\neq k\end{array}}^N\prod_{i=1}^{N}\frac{R_i}{R_j\cdot R_k} = \frac{R_1\cdot R_2 \cdot \ldots \cdot R_{N-1}\cdot R_N}{R_1\cdot R_2 + \ldots + R_{N-1}\cdot R_N}$$

 

[Ben Niehoff]

For resistors specifically, engineers often introduce a new operator || ("parallel") as a shorthand, defined by

A || B = AB/(A + B)

Therefore, for several resistors, one might write

R1 || R2 || R3 || ... || Rn

This operator is commutative and associative, so no need to write parentheses.

But of course, it won't make sense outside the context of circuits (and with resistors specifically, or possibly complex impedances).

Also, Hootenanny's formula is incorrect. What you want is

$$R_1 || R_2 || ... || R_N = \frac{\prod_{i=1}^N R_i}{\sum_{j=1}^N \left( \frac{1}{R_j} \prod_{k=1}^N R_k \right) }$$

For the more general case (for example, in polynomials, where you might want all the products of k objects out of n), then you need to define some "index function" that selects each combination of k indices out of n. I don't know of an official way to notate this, so you can make something up. Greek letters are good: pi is used in this sense to denote permutations, and sigma is used to denote the parity of a permutation. So maybe kappa would be good for combinations? Then you could write

$$\sum_{\kappa(k, n)} \prod_{i \in \kappa} a_i$$

This is a bit abstract, but the point is that the reader can tell what you mean.