Change of variables (i don,t understand)

[eljose]

let be the integral:

\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx

i make the change of variable xy=u y=v whose Jacobian is 1/v but then what would be the new limits?...

 

[OlderDan]

let be the integral:

\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx

i make the change of variable xy=u y=v whose Jacobian is 1/v but then what would be the new limits?...

v is y, so u = xv, and x can never be less than 1. What does that tell you about the possible values of u for any given v?

 

[eljose]

could you write the new limits...i can,t work it out the new values of v for v gives me (-8,8) (here 8 means infinite but for u i got... (0,8) is that true?..what would happen if y choose the change of variable x/y=u y=v? thanx

 

[OlderDan]

could you write the new limits...i can,t work it out the new values of v for v gives me (-8,8) (here 8 means infinite but for u i got... (0,8) is that true?..

From your original integral

\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx

you have y ranging over all reals and x ranging from +1 to infinity. With v = y, v will range over all reals and with u = xy = xv, u will range from -infinity to v when v is negative and from v to infinity when v is positive. Looks like that gives you

\int_{-\infty}^{0}\int_{-\infty}^{v}g(u,v)dudv + \int_{0}^{\infty}\int_v^{\infty}g(u,v)dudv

g(u,v) includes the contribution from f(x,y) and the Jacobian, and you need to be careful with the signs.