I'm trying to solve summation over a matrix that has been populated with an equation using two variables.
Trying to derive the a matrix populated with the equation:
$$f(x,y) = \frac{a}{x^2 + y - b}$$
where $a$ and $b$ are constants.
now given $\max(x) = X$ and $\max(y)=Y$, what is the sum of all the items in the matrix? ie
$$\operatorname{total}(X,Y) = \sum_{x=1}^{X} \sum_{y=1}^{Y}f(x,y)$$
thus, I am trying to derive the $$ \operatorname{total(X,Y)} $$ function, which will give a total of a matrix with entries populated using the above equation.
I think I need split up: $$\operatorname{total}(X,Y) = \sum_{x=1}^{X} \sum_{y=1}^{Y} f(x,y)$$
i.e. solve for inner sigma, finding:
$$g(x; Y)=\sum_{y=1}^{Y} \frac{a}{x^2 + y - b}$$
giving equation $g(x; Y)$, then
$$\operatorname{total}(X,Y) = \sum_{x=1}^{X} g(x; Y)$$
However, I don't know if I'm right on those steps, and I'm not sure how to find the partial sum equation for both steps.
Any math wizz's able to give some suggestions?
Many thanks.