Is it true that Hermite’s functions are bounded for any $n\in\mathbb{N_0},x\in \mathbb{R}?$ If it's true, how to prove it??
2026-03-25 00:06:36.1774397196
Boundedness of the Hermite functions.
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Of course they are not. Polynomials never are. This is because as the argument (say $x$) gets large, the main term dominates all other terms (in the sense that it is a multiple of at least $x$ of any other term), so on letting $x \to \infty$ the polynomial will converge to $+\infty$ or $-\infty$ according to the sign of the main term.
For the physicist's Hermite functions, please see the comments.