  This is the first part of a paper that studies the phase transition in the
asymptotic limit of the rank 1 real Wishart spiked model. In this paper, we
consider $N$-dimensional real Wishart matrices $S$ in the class
$W_{\mathbb{R}}\left(\Sigma,M\right)$ in which all but one eigenvalues of
$\Sigma$ is $1$. Let the non-trivial eigenvalue of $\Sigma$ be $1+\tau$, then
as $N$, $M\rightarrow\infty$, with $N/M=\gamma^2$ finite and non-zero, the
eigenvalue distribution of $S$ will converge into the Machenko-Pastur
distribution inside a bulk region. As $\tau$ increases from zero, one starts
seeing stray eigenvalues of $S$ outside of the support of the Machenko-Pastur
density. As the first of these stray eigenvalues leaves the bulk region, a
phase transition will occur in the largest eigenvalue distribution of the
Wishart matrix. In this paper will compute the asymptotics of the largest
eigenvalue distribution when the phase transition occur. In the this first half
of the paper, we will establish the results that are valid for all $N$ and $M$
and will use them to carry out the asymptotic analysis in the second half of
the paper, which will follow shortly. In particular, we have derived a formula
for the integral $\int_{O(N)}e^{-\tr(XgYg^T)}g^T\D g$ when $X$, $Y$ are
symmetric and $Y$ is a rank 1 matrix. This allows us to write down a Fredholm
determinant formula for the largest eigenvalue distribution and analyze it
using orthogonal polynomial techniques. This approach is very different from a
recent paper by Bloemendal and Virag, in which the largest eigenvalue
distribution was obtained using stochastic operator method.
