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Give a formula the open unit disk

  • Thread starter nhartung
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  • #1
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Homework Statement



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

(a) [tex]\frac{\partial \sum ^{inf}_{n=0} (xy) ^{n}}{\partial x}[/tex]

(b) [tex]\frac{\partial (e^{x+y} \int ^{y}_{x} e^{-t} dt)}{\partial y}[/tex]

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 [tex]\sum ^{inf}_{n=0} (xy)^{n} = \frac{1}{1 - xy}[/tex] if xy < 1 (But for all (x,y)[tex]\in[/tex] D xy < 1

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

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
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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.
 

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