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theorem

Gaussian functions are integrable

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.

theorem

Gaussian functions are almost everywhere strongly measurable

#gaussian_aestronglyMeasurable

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 $\mathbb{R}$ .

theorem

Gaussian Functions Belong to the Hilbert Space

#gaussian_memHS

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 $L^2(\mathbb{R})$ ).

theorem

Integrability of $e^{cx - bx^2}$ for $b > 0$

#exp_mul_gaussian_integrable

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.

theorem

Integrability of $e^{|cx|} e^{-bx^2}$ for $b > 0$

#exp_abs_mul_gaussian_integrable

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.

theorem

$f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^1$

#mul_gaussian_mem_Lp_one

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.

theorem

$f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^2$

#mul_gaussian_mem_Lp_two

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 $L^2(\mathbb{R})$ .

theorem

$f \in \mathcal{H} \implies |f(x)| e^{-b(x-c)^2} \in L^1$

#abs_mul_gaussian_integrable

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.

theorem

$f \in \mathcal{H} \implies e^{cx} |f(x)| e^{-bx^2} \in L^1$

#exp_mul_abs_mul_gaussian_integrable

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.

theorem

$f \in \mathcal{H} \implies e^{|cx|} |f(x)| e^{-bx^2} \in L^1$

#exp_abs_mul_abs_mul_gaussian_integrable

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.

Physlib.QuantumMechanics.OneDimension.HilbertSpace.Gaussians

Gaussian functions are integrable

Gaussian functions are almost everywhere strongly measurable

Gaussian Functions Belong to the Hilbert Space

Integrability of ecx−bx2e^{cx - bx^2}ecx−bx2 for b>0b > 0b>0

Integrability of e∣cx∣e−bx2e^{|cx|} e^{-bx^2}e∣cx∣e−bx2 for b>0b > 0b>0

f∈H ⟹ f(x)e−b(x−c)2∈L1f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^1f∈H⟹f(x)e−b(x−c)2∈L1

f∈H ⟹ f(x)e−b(x−c)2∈L2f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^2f∈H⟹f(x)e−b(x−c)2∈L2

f∈H ⟹ ∣f(x)∣e−b(x−c)2∈L1f \in \mathcal{H} \implies |f(x)| e^{-b(x-c)^2} \in L^1f∈H⟹∣f(x)∣e−b(x−c)2∈L1

f∈H ⟹ ecx∣f(x)∣e−bx2∈L1f \in \mathcal{H} \implies e^{cx} |f(x)| e^{-bx^2} \in L^1f∈H⟹ecx∣f(x)∣e−bx2∈L1

f∈H ⟹ e∣cx∣∣f(x)∣e−bx2∈L1f \in \mathcal{H} \implies e^{|cx|} |f(x)| e^{-bx^2} \in L^1f∈H⟹e∣cx∣∣f(x)∣e−bx2∈L1

Integrability of $e^{cx - bx^2}$ for $b > 0$

Integrability of $e^{|cx|} e^{-bx^2}$ for $b > 0$

$f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^1$

$f \in \mathcal{H} \implies f(x) e^{-b(x-c)^2} \in L^2$

$f \in \mathcal{H} \implies |f(x)| e^{-b(x-c)^2} \in L^1$

$f \in \mathcal{H} \implies e^{cx} |f(x)| e^{-bx^2} \in L^1$

$f \in \mathcal{H} \implies e^{|cx|} |f(x)| e^{-bx^2} \in L^1$