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1: 1.16 Distributions

… ►The linear space of all test functions with the above definition of convergence is called a test function space. … ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►

1.16.21 $$x_{+}^{\alpha }=\frac{1}{{\left(\alpha +1\right)}_n}{𝐷}^n{x}_{+}^{\alpha +n},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $𝐷f$: distributional derivative

Permalink:

http://dlmf.nist.gov/1.16.E21

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §1.16(iv), §1.16 and Ch.1

… ►(See the definition of a distribution in §1.16(i).) … ►

1.16.35 $$⟨ℱ\left(u\right),\varphi ⟩=⟨u,ℱ(\varphi )⟩,$$ $\varphi \in {𝒯}_n$.

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Defines:

$ℱ\left(u\right)$: Fourier transform of a tempered distribution

Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution, $\in$: element of, $n$: nonnegative integer, $𝒯$: space of test functions, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function and $ℱ(\varphi )$: Fourier transform of a test function

Referenced by:

(1.16.36), (1.16.37), item (i), §1.16(viii), item (i)

Permalink:

http://dlmf.nist.gov/1.16.E35

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$⟨ℱ\left(u\right),\varphi ⟩=⟨u,F⟩,$$ $\varphi \in {𝒯}_n$,

where $F(x)$ was the local definition of the Fourier transform in (1.16.29).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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2: 26.2 Basic Definitions

§26.2 Basic Definitions

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Permutation

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Lattice Path

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Partition

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3: 7.2 Definitions

§7.2 Definitions

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§7.2(i) Error Functions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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§7.2(v) Goodwin–Staton Integral

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4: 4.2 Definitions

§4.2 Definitions

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§4.2(i) The Logarithm

… ►With this definition the general logarithm is given by … ►We regard this as the closed definition of the principal value. … ►In contrast to (4.2.5) the closed definition is symmetric. …

5: Need Help?

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6: 4.14 Definitions and Periodicity

§4.14 Definitions and Periodicity

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7: 34.6 Definition: $\mathit{9}j$ Symbol

§34.6 Definition: $\mathit{9}j$ Symbol

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8: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►(Sixty-six equivalent definitions of $C\left(n\right)$ are given in Stanley (1999, pp. 219–229).) …

9: 18.3 Definitions

§18.3 Definitions

►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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10: 5.2 Definitions

§5.2 Definitions

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Euler’s Integral

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