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11: 24.19 Methods of Computation

… ►If ${\tilde{N}}_{2n}$ denotes the right-hand side of (24.19.1) but with the second product taken only for $p\le \lfloor {(\pi \mathrm{e})}^{-1}2n\rfloor +1$, then $N_{2n}=\lceil {\tilde{N}}_{2n}\rceil$ for $n\ge 2$. …

12: 3.2 Linear Algebra

… ►

§3.2(iv) Eigenvalues and Eigenvectors

►If $𝐀$ is an $n\times n$ matrix, then a real or complex number $\lambda$ is called an eigenvalue of $𝐀$, and a nonzero vector $𝐱$ a corresponding (right) eigenvector, if … ►where $𝐱$ and $𝐲$ are the normalized right and left eigenvectors of $𝐀$ corresponding to the eigenvalue $\lambda$. …When $𝐀$ is a symmetric matrix, the left and right eigenvectors coincide, yielding $\kappa (\lambda )=1$, and the calculation of its eigenvalues is a well-conditioned problem. …

13: 19.21 Connection Formulas

… ►

19.21.7 $$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3y^{1/2}{x}^{-1/2}{z}^{-1/2},$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.10 $$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+x^{1/2}{y}^{1/2}{z}^{-1/2},$$ $z\ne 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)

Permalink:

http://dlmf.nist.gov/19.21.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

An incorrect factor of 3 was removed in the last term on the right-hand side.

Suggested 2022-06-27 by Abdulhafeez Abdulsalam

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

►Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)). …

14: 19.25 Relations to Other Functions

… ►

19.25.7 $$E(\varphi ,k)=2R_G(c-1,c-k^2,c)-(c-1)R_F(c-1,c-k^2,c)-\sqrt{c-1}\sqrt{c-k^2}/\sqrt{c},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

… ►All terms on the right-hand sides are nonnegative when $k^2\le 0$, $0\le k^2\le 1$, or $1\le k^2\le c$, respectively. … ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). … ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which , for some $j$. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which , for some $j$. …

15: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

… ►When the series on the right-hand side converges in the half-plane . …

16: 1.6 Vectors and Vector-Valued Functions

… ►where $𝐧$ is the unit vector normal to $𝐚$ and $𝐛$ whose direction is determined by the right-hand rule; see Figure 1.6.1. ►

►►

Figure 1.6.1: Vector notation. Right-hand rule for cross products. Magnify

…

17: 2.5 Mellin Transform Methods

… ►The domain of analyticity of $ℳf\left(z\right)$ is usually an infinite strip parallel to the imaginary axis. … ►Similarly, if $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for $I(x)$ for large values of $x$. … ►We now apply (2.5.5) with , and then translate the integration contour to the right. … ►

2.5.19 $$f(t)=O\left(t^{-b}\right),$$ $t\to +\mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $f(x)$: locally integrable function and $b$: right endpoint

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.27 $$-b_s/\left(z-{\beta }_s\right).$$

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Symbols:

$b$: right endpoint

Permalink:

http://dlmf.nist.gov/2.5.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

…

18: 1.8 Fourier Series

… ►at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). … …

19: Guide to Searching the DLMF

… ►The natural thing is to enter the query trigonometric, and expect to have all the right matches. …

20: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …