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11: 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). …

12: 10.34 Analytic Continuation

… ►For complex $\nu$ replace $\nu$ by $\overline{\nu }$ on the right-hand sides.

13: 18.7 Interrelations and Limit Relations

… ►

18.7.5 $$V_n\left(x\right)=P_n^{(-\frac{1}{2},\frac{1}{2})}\left(x\right)/P_n^{(-\frac{1}{2},\frac{1}{2})}\left(1\right),$$

ⓘ

Symbols:

$W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(viii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E5

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata.

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.6 $$W_n\left(x\right)=(2n+1)P_n^{(\frac{1}{2},-\frac{1}{2})}\left(x\right)/P_n^{(\frac{1}{2},-\frac{1}{2})}\left(1\right).$$

ⓘ

Symbols:

$W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(i), §18.3, §18.5(iii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata.

See also:

Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.17 $$U_{2n}\left(x\right)=W_n\left(2x^2-1\right),$$

ⓘ

Symbols:

$W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.30

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E17

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $V_n\left(2x^2-1\right)$, with $W_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

►

18.7.18 $$T_{2n+1}\left(x\right)=x{V}_n\left(2x^2-1\right).$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.29

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E18

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $x{W}_n\left(2x^2-1\right)$, with $x{V}_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

…

14: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Referenced by:

Erratum (V1.0.5) for Equation (21.3.4)

Permalink:

http://dlmf.nist.gov/21.3.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the vector ${𝐦}_2$ on the right-hand-side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

…

15: 28.26 Asymptotic Approximations for Large $q$

… ►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …

16: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

…

17: 11.9 Lommel Functions

… ►the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer. … ►If either of $\mu \pm \nu$ equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents $S_{\mu ,\nu }\left(z\right)$ exactly. …

18: 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. …

19: 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. …

20: 4.40 Integrals

… ►For the right-hand side see (4.23.39) and (4.23.40). …