= Smooth-data instability of the Cauchy-Riemann Cauchy problem
{title2=$\phi_n(x,t)=e^{-\sqrt n}\cos(n(x+it))$}
For the <first-order Cauchy-Riemann operator>, the equation $\phi_t-i\phi_x=0$ admits the <entire functions> $\phi_n(x,t)=e^{-\sqrt n}\cos(n(x+it))$. Every fixed-order <derivative> of the initial data converges uniformly to zero, since $n^r e^{-\sqrt n}\to0$. Nevertheless $\sup_x|\phi_n(x,t)|=e^{-\sqrt n}\cosh(n|t|)\to\infty$ for each $t\ne0$. Thus even convergence of all initial <derivatives> does not give <continuous dependence on initial data> away from the initial line. Analytic existence and smooth-data stability are different properties.
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