= Fredholm determinant
{c}
{title2=$\det(I+K)=\prod_j(1+\lambda_j)$}
For a trace-class operator $K$ on a <Hilbert space>, the Fredholm determinant is the convergent product over its eigenvalues, counted with algebraic multiplicity. For self-adjoint $K$, <spectral theorem for compact self-adjoint operators> reduces the product to a diagonal calculation. For $\|K\|<1$, $\log\det(I+K)=\sum_{n\geq1}(-1)^{n+1}\operatorname{tr}(K^n)/n$. This identity exposes the successive <cumulants> in a <Gaussian quadratic exponential moment>.
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