Summation with exclusion or skipped step

Question

For example, when $i=0$ and $j=0$, the term $\displaystyle\frac{1}{i+j}$ is invalid. How do I indicate in the equation that $i$ and $j$ cannot be $0$ at the same time and this case should be excluded from the summation? $$\sum_{i=-5}^{5} \sum_{j=-5}^{5}{\frac{1}{i+j}}$$

Answer 1

The Iverson bracket does exactly what you've asked for: $$\sum_{i=-5}^5 \sum_{j=-5}^5 \frac{[i+j \not=0]}{i+j}$$ For summation, the convention is that if the Iverson bracket returns false, the term is omitted.

But if you don't like the Iverson bracket, here are a few other ways to exclude the cases with a zero denominator:

$$\sum_{i=-5}^5 \sum_{j\in\{-5,\dots,5\}\setminus\{-i\}}\frac{1}{i+j}$$

$$\sum_{\substack{-5 \le i, j \le 5:\\ i+j \not= 0}}\frac{1}{i+j}$$

$$\sum_{\substack{(i,j)\in \{-5,\dots,5\} \times \{-5,\dots,5\}:\\ i+j \not= 0}}\frac{1}{i+j}$$

$$\sum_{\substack{(i,j)\in \{-5,\dots,5\}^2:\\ i+j \not= 0}}\frac{1}{i+j}$$

Answer 2

First, why haven't you mentioned the case where $i=1,j=-1$?

But glossing over that, I could swear I've seen people put a prime symbol on the sigma, $\sum'$, to indicate "please skip index values that make the term nonsensical", but I can neither find a reference for this, or figure out exactly how you'd do it with your double summation symbols. Maybe

$$\sum_{i,j=-5}^{5}{\vphantom{\sum}}' \frac{1}{i+j}$$

or

$$\sum_{-5 \le i,j \le 5}{\vphantom{\sum}}' \frac{1}{i+j}$$

Regardless, you should still explain what you mean by the primed sigma, as I don't think it's a standard notation.

Further googling uncovers that the "primed sigma" notation is used when summing over a lattice, when studying doubly periodic complex functions a la Weierstrass, and means "skip the point $(0,0)$" when used there. So not the exact same thing, but possibly close enough to validate borrowing it for your purpose.

Answer 3

Another common notation is \begin{align*} \sum_{i=-5}^5\sum_{\color{blue}{{j=-5}\atop{j\ne -i}}}^{5}\frac{1}{i+j} \end{align*}