Extended Canonical Transformation

[Magister]

Homework Statement

The transformation equations are:

$$Q=q^\alpha cos(\beta p)$$

$$P=q^\alpha sin(\beta p)$$

For what values of $\alpha$ and $\beta$ do these equations represent an extended canonical transformation?

The Attempt at a Solution

Well, just for a start, what is the condition for a canonical transformation to be an extended canonical transformation?

 

[Magister]

I got a solution but it doesn't seems very satisfactory

I believe that the condition for a canonical transformation to be an extended canonical transformation is that

$$PQ^\prime = \lambda p q^\prime$$

But I am not 100% sure.

Then I do

$$PQ^\prime = q^\alpha sen(\beta p)(\alpha q^{\alpha-1}q^\prime cos(\beta p)-q^\alpha \beta sen(\beta p) p^\prime)$$

Now I do the small angle approximation saying that $\beta$ is small. Is this point that I am not sure because the problem statement don't gives any information about this approximation.
However, doing this I get:

$$PQ^\prime \simeq \alpha q^{2\alpha -1} \beta p q^\prime - \beta^3 q^{2\alpha} p^2 p^\prime$$

Using

$$\beta^3 \simeq 0$$

and

$$\alpha=\frac{1}{2}$$

I get

$$PQ^\prime \simeq \frac{\beta}{2} p q^\prime$$

At least I get the condition of an extended canonical transformation

Am I thinking right?
Thanks for any suggestion.