# Series & Integrals 2

1. May 14, 2009

### joemama69

1. The problem statement, all variables and given/known data

show what integral of 1/(1-xy)dxdy = pi2/6

dx 0 to 1
dy 0 to 1

2. Relevant equations

3. The attempt at a solution

2. May 14, 2009

### Staff: Mentor

Doesn't look like a hard integral. What have you tried?

3. May 14, 2009

### joemama69

$$\int$$1/(1-xy)dx = ln(1-xy) = ln(1-y) - ln1

$$\int$$ln(1-y) - ln1 dy = (1-y)ln(1-y) - (1- y) - yln1

= -ln1 - (ln1 - 1) = = -2ln1 + 1 huh

4. May 14, 2009

### Firepanda

remember

$$\int$$f '(x)/f(x) dx = ln[f(x)] for your 1st integral, also you're taking y constant, so treat it like one

5. May 14, 2009

### Count Iblis

1/(1-xy) = sum from n= 0 to infinity of x^n y^n

The integral is thus the sum from n = 0 to infinity of 1/(n+1)^2 which is pi^2/6

6. May 14, 2009

### Cyosis

The first integration isn't correct. You can see that immediately by differentiating the primitive. Use a substitution if you don't see it right away, $u=1-xy,du=-ydx$. The hard part comes after this however. Hint: Use the power series of the logarithm.

Edit: Use Count's method it's quicker and easier.

7. May 15, 2009

### joemama69

1(1-xy) = $$\sum$$xnyn from n = 0 to infinity

$$\int\int$$1/(1-xy) = $$\sum$$1/(n+1)2 = pi2/6

how does 1/(1-xy) go to the sum of xnyn

and how does 1/(1-xy) got to the sum of (n+1)2 then go to pi2/2

8. May 15, 2009

### Cyosis

Ok let's say z=xy, then $$\frac{1}{1-xy}= \frac{1}{1-z}=\sum_{n=0}^\infty z^n=\sum_{n=0}^\infty (xy)^n=\sum_{n=0}^\infty x^n y^n$$. Then you take the integral over the sum.

$$\int_0^1 \int_0^1 \sum_{n=0}^\infty x^n y^n dxdy=\sum_{n=0}^\infty \left(\int_0^1 \int_0^1 x^n y^n dxdy\right)$$. Can you see how to continue from here?

Edit: I think you're expected to know that the series converges to pi^2/6 by heart. However http://en.wikipedia.org/wiki/Basel_problem shows you how to get to that value in case you're interested.

Last edited: May 15, 2009
9. May 15, 2009

### joemama69

oops i think i forgot something

the entire problem is

show that

$$\int\int$$1/(1-xy)dxdy = pi2/6

by doing the double substitution

x = (u - v)/$$\sqrt{2}$$ and y = (u + v)/$$\sqrt{2}$$

This amounts to rotating the axes cuonterclockwise through the angle pi/4