Give a formula the open unit disk

[nhartung]

Homework Statement

In the open unit disk D = {(x,y)|x² + y² < 1} give a formula for:

(a) $$\frac{\partial \sum ^{inf}_{n=0} (xy) ^{n}}{\partial x}$$

(b) $$\frac{\partial (e^{x+y} \int ^{y}_{x} e^{-t} dt)}{\partial y}$$

Homework Equations

The Attempt at a Solution

Ok this is on my exam review sheet and he gave us the solutions to go along with it.. I don't know if I wasn't paying attention in class or what but I don't remember doing anything like this.

His first step for a is $$\sum ^{inf}_{n=0} (xy)^{n} = \frac{1}{1 - xy}$$ if xy < 1 (But for all (x,y)$$\in$$ D xy < 1

I'm already confused at this point. Can someone please let me know what is going on here? Thanks

 

[Dick]

The sum of r^n is 1/(1-r) if |r|<1. Now put r=(xy). It's just a geometric series. For the second one just evaluate the integral and take the partial derivative using the product rule etc. There's nothing really special going on there.