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1: Browsers

… ►Although we have attempted to follow standards and maintain backwards compatibility with older browsers, you will generally get the best results by upgrading to the latest version of your preferred browser.

2: Guide to Searching the DLMF

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$	stands for any number of alphanumeric characters
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	(If ? occurs as a separate token, it effectively acts like $)

… ►Be careful, however, because if you put quotes around math expressions involving math symbols, you may not get the matches you’d expect. … ►

Metadata for More Effective Search

►To find more effectively the information you need, especially equations, you may at times wish to specify what you want with descriptive words that characterize the contents but do not occur literally. …

3: 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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4: 4.10 Integrals

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5: 19.22 Quadratic Transformations

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19.22.5 $$2p_{\pm }=\sqrt{(p+x)(p+y)}\pm \sqrt{(p-x)(p-y)},$$

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Defines:

$p_{\pm }$: modified parameter (locally)

Referenced by:

§19.22(iii), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.22(i), §19.22 and Ch.19

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$4(p_{\pm }^2-a^2)={(\sqrt{p^2-x^2}\pm \sqrt{p^2-y^2})}^2.$

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$2z_{\pm }=\sqrt{(z+x)(z+y)}\pm \sqrt{(z-x)(z-y)},$

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$4(z_{\pm }^2-a^2)={(\sqrt{z^2-x^2}\pm \sqrt{z^2-y^2})}^2.$

… ►However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. …

6: 1.11 Zeros of Polynomials

… ►Add $-\frac{1}{4}a$ to the roots of $g(w)=0$ to get those of $f(z)=0$. …

7: Foreword

… ►The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. The DLMF may well serve as a model for the effective presentation of highly mathematical reference material on the Web. …

8: 19.25 Relations to Other Functions

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

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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

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19.25.36 $$\mathrm{\wp }\left(z\right)-e_j\in ℂ\setminus (-\mathrm{\infty },0],$$ $j=1,2,3.$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $j$: $(=1,2,3)$ and $z$: complex number

Referenced by:

(19.25.39)

Permalink:

http://dlmf.nist.gov/19.25.E36

Encodings:

pMML, png, TeX

See also:

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

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

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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

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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

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9: Frank W. J. Olver

… ►He also spent time as a Visiting Fellow, or Professor, at the University of Lancaster, U. … ►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). …

10: Possible Errors in DLMF

… ►Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …