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21: 12.12 Integrals

… ►When $z$ $(=x)$ is real the left-hand side equals ${(F(a,x))}^2$; compare (12.2.22). …

22: 14.24 Analytic Continuation

… ►The behavior of $P_{\nu }^{-\mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ as $z\to -1$ from the left on the upper or lower side of the cut from $-\mathrm{\infty }$ to $1$ can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with $s=\pm 1$.

23: 26.13 Permutations: Cycle Notation

… ►A permutation that consists of a single cycle of length $k$ can be written as the composition of $k-1$ two-cycles (read from right to left): …

24: 27.4 Euler Products and Dirichlet Series

… ►if the series on the left is absolutely convergent. …

25: 19.25 Relations to Other Functions

… ►

19.25.35 $$z+2\omega =\pm R_F(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots and $z$: complex number

Proof sketch:

Use (23.6.36) with $z=\mathrm{\wp }\left(\omega \right)$ as prescribed in the text that follows (23.6.36), substitute $u=t+\mathrm{\wp }\left(\omega \right)$ and compare with (19.16.1). The undetermined $\pm$ is a consequence of the multivaluedness of the square-roots in (19.16.4).

Referenced by:

(19.25.38), (19.25.40), §19.25(v), §19.25(vi), §19.25(vi), §20.9(i), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E35

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

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

… ►

19.25.37 $$\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)=\pm 2R_G(\mathrm{\wp }\left(z\right)-e_1,\mathrm{\wp }\left(z\right)-e_2,\mathrm{\wp }\left(z\right)-e_3),$$

ⓘ

Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function and $z$: complex number

Referenced by:

§19.25(vi), Erratum (V1.0.18) for Equation (19.25.37), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E37

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $\zeta \left(z\right)+z\mathrm{\wp }\left(z\right)$, has been replaced by $\zeta \left(z+2\omega \right)+(z+2\omega )\mathrm{\wp }\left(z\right)$, and the right-hand side has been multiplied by $\pm 1$.

Clarification (effective with 1.0.18):

The Weierstrass zeta function in this equation incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the functions appeared correct.

See also:

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

… ►

19.25.39 $$\zeta \left({\omega }_j\right)+{\omega }_j{e}_j=2R_G(0,e_j-e_k,e_j-e_{\mathrm{\ell }}),$$

ⓘ

Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $e_j$: Weierstrass lattice roots, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $j$: $(=1,2,3)$ , $z$: complex number, $k$: $(=1,2,3)$ , $\mathrm{\ell }$: $(=1,2,3)$ , ${\omega }_j$: nonzero complex number and ${\eta }_j$: complex number

Proof sketch:

Take $z=-\omega$ in (19.25.36).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E39

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected such that the left-hand side ${\eta }_j$ has been replaced by $\zeta \left({\omega }_j\right)$.

See also:

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

… ►

19.25.40 $$z+2\omega =\pm \sigma \left(z\right)R_F({\sigma }_1^2(z),{\sigma }_2^2(z),{\sigma }_3^2(z)),$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $z$: complex number and ${\sigma }_j(z)$: function

Proof sketch:

Combine Erdélyi et al. (1953b, §§13.12(22), 13.13(22)), (19.25.35) and (19.16.1).

Referenced by:

§19.25(vi), §19.25(vi), Erratum (V1.1.0) for Subsection 19.25(vi)

Permalink:

http://dlmf.nist.gov/19.25.E40

Encodings:

pMML, png, TeX

Correction (effective with 1.1.0):

This equation has been corrected, namely the left-hand side which was originally $z$, has been replaced by $z+2\omega$ and the right-hand side has been multiplied by $\pm 1$.

See also:

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

…

26: 1.4 Calculus of One Variable

… ►

§1.4(ii) Continuity

… ►Similarly, it is continuous on the left (or from below) at $x=c$ if ►

1.4.3 $$f(c-)\equiv \underset{x\to c-}{lim}f(x)=f(c).$$

ⓘ

Symbols:

$\equiv$: equals by definition

Referenced by:

§1.4(ii)

Permalink:

http://dlmf.nist.gov/1.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(ii), §1.4 and Ch.1

… ►If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$. …

27: 4.13 Lambert $W$-Function

… ►

►►

Figure 4.13.2: The $W\left(z\right)$ function on the first 5 Riemann sheets. $W\left(z\right)$ maps the first Riemann sheet in the middle of the left-hand side to the region enclosed by the green curve on the right-hand side; it maps the Riemann sheet on the left-hand side to the region enclosed by the pink, green and orange curves on the right-hand side, etc. Magnify

… ►For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. …

28: 4.24 Inverse Trigonometric Functions: Further Properties

… ►The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. …

29: 7.6 Series Expansions

…

30: 15.16 Products

…