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The identity theorem says analytic functions agreeing on a set with an interior accumulation point agree throughout a connected domain. Analytic continuation is an analytic function on a larger connected domain agreeing on a nonempty overlap. Factor the denominator as $(t+i)(t-z)$. For $\Im z<0$, contour closure below giveswith a removable value at $-i$; this entire expression is the continuation. For $\Im z>0$, the displayed real-axis integral encloses only $-i$ and equals $2\pi i{e}^{-5}/(z+i)$, so it is not that continuation. Finally $1/n\to 0$ only at a boundary point of $D$, so the zeros do not determine $G$; for example $G(z)=\sin (\pi /z)$ is nonzero and has them all.

Scaling by $K$, the positive fixed point is $x=1$. Linearization $x_t=1+{ϵ}_t$ gives ${ϵ}_{t+1}={ϵ}_t-r{ϵ}_{t-1}$, with characteristic polynomial ${\lambda }^2-\lambda +r$. The Jury conditions put both roots inside the unit circle exactly for $0<r<1$. At $r=1$ the roots are $e^{\pm i\pi /3}$, sixth roots of unity. The discrete Hopf resonance therefore creates a small period-six branch; expansion of the nonlinear recurrence on the centre eigenspace supplies the nonzero amplitude.

Write $f(y;\theta )=\exp \{y\theta -b(\theta )+c(y)\}$ and use canonical parameter ${\theta }_i=X_i^T\beta$. Thenwhere ${\mu }_i=b^{\prime }(X_i^T\beta )$. Geometrically the residual vector is orthogonal to every design column. For $H_0:{\beta }_1=0$, use a two-sided Wald statistic ${\hat{\beta }}_1/\mathrm{se}({\hat{\beta }}_1)$ (or likelihood-ratio statistic), rejecting at the corresponding normal or chi-square critical value.