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Math Help - Need help with algebra within these integrals

  1. #1
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    Need help with algebra within these integrals

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

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

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

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

    Why is that?
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  2. #2
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    Re: Need help with algebra within these integrals

    Quote Originally Posted by tttcomrader View Post
    I'm working on some integrals of exponential families, but I'm just not sure how the textbook have the followings:

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

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

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

    Why is that?
    1. Integrate wrt x (simple substitution technique - Complete the square, make the substitution, recognise a standard definite integral).

    2. Convergence is required.
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