indices differing by an integer

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11: 18.39 Applications in the Physical Sciences

… ►

An Introductory Remark

… ►The

\epsilon_{n}

are the observable energies of the system, and an increasing function of

n

. … ►with an infinite set of orthonormal

L^{2}

eigenfunctions … ►This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed

l

, do not form a complete set. … ►see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an

n!

in the denominator. …

12: 1.2 Elementary Algebra

… ►with matrix elements

a_{ij}\in\mathbb{C}

, where

i

j

are the row and column indices, respectively. … ►Assuming the indicated multiplications are defined: matrix multiplication is associative … ►The

l^{2}

norm is implied unless otherwise indicated. … ►Unless otherwise indicated, matrices are assumed square, of order

n

; and, when vectors are combined with them, these are of length

n

. … ►an Hermitian matrix if …

13: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Then an isomorphism is given by … ►For example, replacing

2q\cos{(2z)}

of (28.2.1) by

\lambda\cos{(2\pi\alpha n+\theta)}

n\in\mathbb{Z}

gives an almost Mathieu equation which for appropriate

\alpha

has such properties. … ►

Spectrum of an Operator

… ►The two (equal) deficiency indices of

\mathcal{L}

are then equal to

n_{1}+n_{2}-2

. … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. …

14: 21.7 Riemann Surfaces

… ►On this surface, we choose

2g

cycles (that is, closed oriented curves, each with at most a finite number of singular points)

a_{j}

b_{j}

j=1,2,\dots,g

, such that their intersection indices satisfy … ►Next, define an isomorphism

\boldsymbol{{\eta}}

which maps every subset

T

B

with an even number of elements to a

2g

-dimensional vector

\boldsymbol{{\eta}}(T)

with elements either

0

\tfrac{1}{2}

. … ►

21.7.12

T_{1}\ominus T_{2}=(T_{1}\cup T_{2})\setminus(T_{1}\cap T_{2}).

ⓘ

Symbols:: $\cap$ : intersection, $\setminus$ : set subtraction, $\cup$ : union and $T$ : subset of $B$
Permalink:: http://dlmf.nist.gov/21.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(iii), §21.7 and Ch.21

… ►

21.7.14

\boldsymbol{{\eta}}(T_{1}\ominus T_{2})=\boldsymbol{{\eta}}(T_{1})+\boldsymbol% {{\eta}}(T_{2}),

ⓘ

Symbols:: $T$ : subset of $B$
Permalink:: http://dlmf.nist.gov/21.7.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(iii), §21.7 and Ch.21

►

21.7.15

4\boldsymbol{{\eta}}^{1}(T)\cdot\boldsymbol{{\eta}}^{2}(T)=\tfrac{1}{2}\left(|% T\ominus U|-g-1\right)\pmod{2},

ⓘ

Symbols:: $g$ : positive integer, $U$ : subset of $B$ and $T$ : subset of $B$
Permalink:: http://dlmf.nist.gov/21.7.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(iii), §21.7 and Ch.21

…

15: Errata

… ►

Section 16.11(i)

A sentence indicating that explicit representations for the coefficients $c_{k}$ are given in Volkmer (2023) was inserted just below (16.11.5).

… ►

Subsection 14.6(ii)

A sentence was added at the end of the subsection indicating that for generalizations, see Cohl and Costas-Santos (2020).

… ►

Section 1.13

In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z)

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$ -argument Wronskian is given by $\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]$ , where $1\leq j,k\leq n$ . Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$ -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$ th-order differential equations. A reference to Ince (1926, §5.2) was added.

… ►

Section 10.8

A sentence was added just below (10.8.3) indicating that it is a rewriting of (16.12.1).

Suggested by Tom Koornwinder.

… ►

References

An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).

…

16: 25.11 Hurwitz Zeta Function

… ►

25.11.4

\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{% s}},

m=1,2,3,\dots

ⓘ

Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: recurrence
Proof sketch:: Derivable from (25.11.1) by comparing its sum over $n=0,\ldots,m-1$ , $m\geq 1$ , with its sum over $n\geq m$ .
Permalink:: http://dlmf.nist.gov/25.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(i), §25.11 and Ch.25

… ►where

h, k

are integers with

1\leq h\leq k

and

n=1,2,3,\dots

. … ►

25.11.33

H_{n}=\sum_{k=1}^{n}k^{-1}.

ⓘ

Defines:: $H_{\NVar{n}}$ : harmonic number
Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonic number
Source:: Knuth (1968, Section 1.2.7, p. 73)
Referenced by:: §25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E33
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonic number $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

… ►

25.11.35

\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\,\mathrm{d}x=2^{-s}\left(% \zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right),

\Re a>0

\Re s>0

; or

\Re a=0

\Im a\neq 0

0<\Re s<1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\int$ : integral, $\Re$ : real part, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: improper integral, infinite series, integral representation, series representation
Proof sketch:: The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$ , factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$ .
Referenced by:: (25.11.31), §25.11(x), §25.11(x)
Permalink:: http://dlmf.nist.gov/25.11.E35
Encodings:: pMML, png, TeX
See also:: Annotations for §25.11(x), §25.11 and Ch.25

… ►For an exponentially-improved form of (25.11.43) see Paris (2005b).