We prove that the set $\{\varphi_0,\varphi_1,\varphi_4,\dots,\varphi_{3k+1},\dots\}$ of Hermite functions is an orthogonal system in the Sobolev space $H^1(\mathbf{R}) = H_{(1)}(\mathbf{R}).$ Furthermore, the logarithmic integral of a function $f$ from the real Hardy space $\mathcal H^1(\mathbf{R})$ is exactly the primitive function of $-\tilde{f}$ (the Hilbert transform of $f$). And more interesting formulas are found for Radon transform of Hermite-like functions.