Physlib.QuantumMechanics.OneDimension.HilbertSpace.Gaussians

For any real numbers $b$ and $c$ such that $b > 0$, the complex-valued function $x \mapsto e^{-b(x-c)^2}$ is integrable on $\mathbb{R}$ with respect to the Lebesgue measure.

For any real numbers $b$ and $c$ such that $b > 0$, the complex-valued function $x \mapsto e^{-b(x-c)^2}$ is almost everywhere strongly measurable with respect to the Lebesgue measure on $\mathbb{R}$.

For any real numbers $b$ and $c$ such that $b > 0$, the complex-valued Gaussian function $x \mapsto e^{-b(x-c)^2}$ is an element of the Hilbert space of square-integrable functions on $\mathbb{R}$ (often denoted as $L^2(\mathbb{R})$).

For any real numbers $b$ and $c$ such that $b > 0$, the function $f(x) = e^{cx} e^{-bx^2}$ is integrable on $\mathbb{R}$ with respect to the Lebesgue measure.

For any real numbers $b$ and $c$ such that $b > 0$, the function $f(x) = e^{|cx|} e^{-bx^2}$ is integrable on $\mathbb{R}$ with respect to the Lebesgue measure.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function in the Hilbert space $\mathcal{H}$ (the space of square-integrable functions $L^2(\mathbb{R})$). For any real numbers $b$ and $c$ such that $b > 0$, the function $x \mapsto f(x) e^{-b(x-c)^2}$ is integrable, i.e., it belongs to $L^1(\mathbb{R})$ with respect to the Lebesgue measure.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function in the Hilbert space $\mathcal{H}$ (the space of square-integrable functions $L^2(\mathbb{R})$). For any real numbers $b$ and $c$ such that $b > 0$, the function $x \mapsto f(x) e^{-b(x-c)^2}$ is square-integrable, i.e., it belongs to $L^2(\mathbb{R})$.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function in the Hilbert space $\mathcal{H}$ (the space of square-integrable functions $L^2(\mathbb{R})$). For any real numbers $b$ and $c$ such that $b > 0$, the function $x \mapsto |f(x)| e^{-b(x-c)^2}$ is integrable with respect to the Lebesgue measure.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function in the Hilbert space $\mathcal{H}$ (the space of square-integrable functions $L^2(\mathbb{R})$). For any real numbers $b$ and $c$ such that $b > 0$, the function $x \mapsto e^{cx} |f(x)| e^{-bx^2}$ is integrable with respect to the Lebesgue measure.

Let $f: \mathbb{R} \to \mathbb{C}$ be a function in the Hilbert space $\mathcal{H}$ (the space of square-integrable functions $L^2(\mathbb{R})$). For any real numbers $b$ and $c$ such that $b > 0$, the function $x \mapsto e^{|cx|} |f(x)| e^{-bx^2}$ is integrable with respect to the Lebesgue measure.