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21: 20.7 Identities

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20.7.5 $${\theta }_3^4(0,q)={\theta }_2^4(0,q)+{\theta }_4^4(0,q).$$

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Symbols:

${\theta }_j(z,q)$: theta function and $q$: nome

A&S Ref:

16.28.5

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/20.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(i), §20.7 and Ch.20

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20.7.9 $${\theta }_4^2(0,q){\theta }_4(w+z,q){\theta }_4(w-z,q)={\theta }_3^2(w,q){\theta }_3^2(z,q)-{\theta }_2^2(w,q){\theta }_2^2(z,q).$$

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Symbols:

${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§20.7(ii)

Permalink:

http://dlmf.nist.gov/20.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(ii), §20.7 and Ch.20

… ►Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices $2,3,4$. … ►See also Carlson (2011, §§1 and 4). … ►

20.7.24 $${\theta }_4\left(4z\right|4\tau )=B{\theta }_4\left(z\right|\tau ){\theta }_4\left(\frac{1}{4}\pi -z\right|\tau ){\theta }_4\left(\frac{1}{4}\pi +z\right|\tau ){\theta }_3\left(z\right|\tau ).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $B$

Permalink:

http://dlmf.nist.gov/20.7.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

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22: Errata

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Version 1.0.25 (December 15, 2019)

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22.9.8 $$s_{1,3}^{(4)}{s}_{2,3}^{(4)}+s_{2,3}^{(4)}{s}_{3,3}^{(4)}+s_{3,3}^{(4)}{s}_{1,3}^{(4)}=\frac{{\kappa }^2-1}{k^2}$$

22.9.9 $$c_{1,3}^{(4)}{c}_{2,3}^{(4)}+c_{2,3}^{(4)}{c}_{3,3}^{(4)}+c_{3,3}^{(4)}{c}_{1,3}^{(4)}=-\frac{\kappa (\kappa +2)}{{(1+\kappa )}^2}$$

22.9.10 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}+d_{2,3}^{(2)}{d}_{3,3}^{(2)}+d_{3,3}^{(2)}{d}_{1,3}^{(2)}=d_{1,3}^{(4)}{d}_{2,3}^{(4)}+d_{2,3}^{(4)}{d}_{3,3}^{(4)}+d_{3,3}^{(4)}{d}_{1,3}^{(4)}=\kappa (\kappa +2)$$

Originally all the functions $s_{m,p}^{(4)}$, $c_{m,p}^{(4)}$ , $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ in Equations (22.9.8), (22.9.9) and (22.9.10) were written incorrectly with $p=2$. These functions have been corrected so that they are written with $p=3$. In the sentence just below (22.9.10), the expression $s_{m,2}^{(4)}{s}_{n,2}^{(4)}$ has been corrected to read $s_{m,p}^{(4)}{s}_{n,p}^{(4)}$.

Reported by Juan Miguel Nieto on 2019-11-07

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Version 1.0.24 (September 15, 2019)

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Version 1.0.23 (June 15, 2019)

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Version 1.0.22 (March 15, 2019)

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23: 12.4 Power-Series Expansions

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12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.5 $$u_1(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(1+(a-\frac{1}{2})\frac{z^2}{2!}+(a-\frac{1}{2})(a-\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_1(a,z)$: solution

A&S Ref:

19.2.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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24: 23 Weierstrass Elliptic and Modular
Functions

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25: 33.25 Approximations

… ►Cody and Hillstrom (1970) provides rational approximations of the phase shift ${\sigma }_0\left(\eta \right)=\mathrm{ph}\mathrm{\Gamma }\left(1+\mathrm{i}\eta \right)$ (see (33.2.10)) for the ranges $0\le \eta \le 2$, $2\le \eta \le 4$, and $4\le \eta \le \mathrm{\infty }$. …09×10⁻²⁰ to 4. …

26: Ranjan Roy

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4 Elementary Functions

… ►In November 2015, Roy was named Associate Editor for Chapters 1 and 4.

27: 12.9 Asymptotic Expansions for Large Variable

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12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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28: 18.4 Graphics

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Figure 18.4.1: Jacobi polynomials $P_n^{(1.5,-0.5)}\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.3: Chebyshev polynomials $T_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.4: Legendre polynomials $P_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.5: Laguerre polynomials $L_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

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Figure 18.4.6: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $\alpha =0,1,2,3,4$. Magnify

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29: 30.16 Methods of Computation

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$$A_{j,j+1}=-{\gamma }^2\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)},$$

… ►For $m=2$, $n=4$, ${\gamma }^2=10$, … ►

${\alpha }_{2,4}=\mathrm{13.97907\hspace{0.33em}459},$

… ►which yields ${\lambda }_4^2\left(10\right)=\mathrm{13.97907\hspace{0.33em}345}$. … ►The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11). …

30: 4.9 Continued Fractions

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4.9.1 $$\ln \left(1+z\right)=\frac{z}{1+}\frac{z}{2+}\frac{z}{3+}\frac{4z}{4+}\frac{4z}{5+}\frac{9z}{6+}\frac{9z}{7+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.39

Permalink:

http://dlmf.nist.gov/4.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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4.9.2 $$\ln \left(\frac{1+z}{1-z}\right)=\frac{2z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\frac{16z^2}{9-}\mathrm{\cdots },$$

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Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.40

Permalink:

http://dlmf.nist.gov/4.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(i), §4.9 and Ch.4

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$$=1+\frac{z}{1-(z/2)+}\frac{z^2/(4\cdot 3)}{1+}\frac{z^2/(4\cdot 15)}{1+}\frac{z^2/(4\cdot 35)}{1+}\mathrm{\cdots }\frac{z^2/(4(4n^2-1))}{1+}\mathrm{\cdots }$$

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4.9.4 $${\mathrm{e}}^z-e_{n-1}(z)=\frac{z^n}{n!-}\frac{n!z}{(n+1)+}\frac{z}{(n+2)-}\frac{(n+1)z}{(n+3)+}\frac{2z}{(n+4)-}\frac{(n+2)z}{(n+5)+}\frac{3z}{(n+6)-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable and $e_n(z)$: expansion of exponential

A&S Ref:

4.2.41

Permalink:

http://dlmf.nist.gov/4.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.9(ii), §4.9 and Ch.4

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