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1: 1.10 Functions of a Complex Variable

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§1.10(xi) Generating Functions

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2: 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, or generalized function, 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 $𝒟(I)$, … ►(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:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: 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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3: 1.13 Differential Equations

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

… ►As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho \in C^2(a,b)$ and $q\in C(a,b)$. …

4: 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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5: 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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6: 5.2 Definitions

§5.2 Definitions

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Euler’s Integral

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7: 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. …

8: Need Help?

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9: 4.14 Definitions and Periodicity

§4.14 Definitions and Periodicity

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10: 34.6 Definition: $\mathit{9}j$ Symbol

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

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