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11: 35.1 Special Notation

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$a,b$	complex variables.
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$\mathrm{d}𝐇$	normalized Haar measure on $𝐎(m)$.
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12: 20.15 Tables

… ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

13: Publications

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Handbook of Mathematical Functions, in A Century of Excellence in Measurements Standards and Technology (D. R. Lide, ed.), CRC Press, pp. 135–139.

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14: 18.33 Polynomials Orthogonal on the Unit Circle

… ►After a quadratic transformation (18.2.23) this would express OP’s on $[-1,1]$ with an even orthogonality measure in terms of the ${\varphi }_n$. … ►Let $\mu$ be a probability measure on the unit circle of which the support is an infinite set. A system of monic polynomials $\{{\mathrm{\Phi }}_n(z)\}$, $n=0,1,\mathrm{\dots }$, where ${\mathrm{\Phi }}_n(x)$ is of proper degree $n$, is orthogonal on the unit circle with respect to the measure $\mu$ if … ►If the measure $\mu$ is absolutely continuous, i. … ►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …

15: 22.18 Mathematical Applications

… ►The arc length $l(u)$ in the first quadrant, measured from $u=0$, is … ►The arc length $l(r)$, measured from $\varphi =0$, is …

16: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$, ►

1.18.11

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.18.E11

Encodings:

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See also:

Annotations for §1.18(ii), §1.18 and Ch.1

… ► $d\alpha (x)=w(x)dx$, see §1.4(v), where the nonnegative weight function $w(x)$ is Lebesgue measurable on $X$. … ►

1.18.13 $$c_n=⟨f,{\varphi }_n⟩={\int }_a^{b}f(x)\overline{{\varphi }_n(x)}dx,$$ $f\in L^2\left(X\right)$,

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Symbols:

$L^2(X,dx)$: Lebesgue–Stieltjes measurable, square integrable, complex-valued functions, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\in$: element of, $⟨f,g⟩$: inner product over functions, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E13

Encodings:

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Annotations for §1.18(ii), §1.18 and Ch.1

… ►For fixed angular momentum $\mathrm{\ell }$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues ${\lambda }_n,n=0,1,\mathrm{\dots },N-1$, with corresponding $L^2([0,\mathrm{\infty }),r^{2}dr)$ eigenfunctions ${\varphi }_n(r)$, and also a continuous spectrum $\lambda \in [0,\mathrm{\infty })$, with Dirac-delta normalized eigenfunctions ${\varphi }_{\lambda }(r)$, also with measure $r^{2}dr$. …

17: 18.39 Applications in the Physical Sciences

… ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure $r^{2}dr\sin \theta d\theta d\varphi$. … ►where the orthogonality measure is now $dr$, $r\in [0,\mathrm{\infty }).$ … ►Orthogonality, with measure $dr$ for $r\in [0,\mathrm{\infty })$, for fixed $l$ … ►thus recapitulating, for $Z=1$, line 11 of Table 18.8.1, now shown with explicit normalization for the measure $dr$. …normalized with measure $r^{2}dr$, $r\in [0,\mathrm{\infty })$. …

18: DLMF Project News

error generating summary

19: 36.7 Zeros

… ►, $y=0$), the number of rings in the $m$th row, measured from the origin and before the transition to hairpins, is given by …

20: 3.2 Linear Algebra

… ►The sensitivity of the solution vector $𝐱$ in (3.2.1) to small perturbations in the matrix $𝐀$ and the vector $𝐛$ is measured by the condition number … ►If $𝐀$ is nondefective and $\lambda$ is a simple zero of $p_n(\lambda )$, then the sensitivity of $\lambda$ to small perturbations in the matrix $𝐀$ is measured by the condition number …