Originally Posted by

**tttcomrader** I'm working on some integrals of exponential families, but I'm just not sure how the textbook have the followings:

1. $\displaystyle \frac {1} {2 \pi } \int \int \exp ( \eta xy - \frac {(x^2+y^2)}{2} ) dxdy $

$\displaystyle = \frac {1}{ \sqrt {2 \pi } } \int \exp [ - \frac {1}{2} (1- \eta ^2)y^2 ] dy $

How does the first line become the second line?

2. Let $\displaystyle A( \eta ) = \log \int ^1 _0 \exp [ \eta \log (x) ] (1-x)^2 dx $

The text says $\displaystyle e ^ { A ( \eta ) } \leq \infty $ only for $\displaystyle \eta \in (-1, \infty ) $

Why is that?