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1: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

… ►

§8.19(ii) Graphics

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§8.19(ix) Inequalities

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§8.19(x) Integrals

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§8.19(xi) Further Generalizations

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2: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ► … ►

Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

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3: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(i) Definitions: General Values

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§8.21(iv) Interrelations

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§8.21(v) Special Values

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4: 1.16 Distributions

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\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}

is called a distribution, or generalized function, 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

\mathcal{D}(I)

, … ►More generally, for

\alpha\colon[a,b]\to[-\infty,\infty]

a nondecreasing function the corresponding Lebesgue–Stieltjes measure

\mu_{\alpha}

(see §1.4(v)) can be considered as a distribution: … ►More generally, if

\alpha(x)

is an infinitely differentiable function, then … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

5: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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Convergence Properties

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§35.8(iv) General Properties

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Confluence

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8: Guide to Searching the DLMF

… ►Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. … ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. …

9: 1.11 Zeros of Polynomials

… ►More generally, for polynomials

f(z)

and

g(z)

, there are polynomials

q(z)

and

r(z)

, found by equating coefficients, such that … ►Next, let

f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dots+a_{0}

. … ►Add

-\tfrac{1}{4}a

to the roots of

g(w)=0

to get those of

f(z)=0

. …

10: 21.1 Special Notation

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$g, h$	positive integers.
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$a\circ b$	intersection index of $a$ and $b$ , two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$ .
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