# definition

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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. … $\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is called a distribution if it is a continuous linear functional on $\mathcal{D}(I)$, that is, it is a linear functional and for every $\phi_{n}\to\phi$ in $\mathcal{D}(I)$, …
1.16.21 $x^{\alpha}_{+}=\frac{1}{{\left(\alpha+1\right)_{n}}}D^{n}x_{+}^{\alpha+n},$
(See the definition of a distribution in §1.16(i).) …
1.16.35 $\left\langle\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle u,% \mathscr{F}(\phi)\right\rangle,$ $\phi\in\mathcal{T}_{n}$.
##### 4: 4.2 Definitions
###### §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. …
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##### 8: 26.5 Lattice Paths: Catalan Numbers
###### §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)). …