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21: 19.16 Definitions

… ►In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. … ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

… ►The $R$-function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 $R_{-a}(𝐛;𝐳)$ was denoted by $R(a;𝐛;𝐳)$. … ►For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b). …

22: 10.36 Other Differential Equations

… ►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

23: 10.43 Integrals

… ►

$$\int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$,

►

$$\int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$$ $\nu \ne \frac{1}{2}$.

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§10.43(ii) Integrals over the Intervals $(0,x)$ and $(x,\mathrm{\infty })$

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§10.43(iv) Integrals over the Interval ($0,\mathrm{\infty }$)

… ► …

24: 4.4 Special Values and Limits

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4.4.2 $$\ln \left(-1\pm \mathrm{i}0\right)=\pm \pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.14 (modified)

Permalink:

http://dlmf.nist.gov/4.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

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4.4.3 $$\ln \left(\pm \mathrm{i}\right)=\pm \frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\ln z$: principal branch of logarithm function

A&S Ref:

4.1.15

Permalink:

http://dlmf.nist.gov/4.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(i), §4.4 and Ch.4

… ►

4.4.6 $${\mathrm{e}}^{\pm \pi \mathrm{i}/2}=\pm \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

A&S Ref:

4.2.27

Permalink:

http://dlmf.nist.gov/4.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.8 $${\mathrm{e}}^{\pm \pi \mathrm{i}/3}=\frac{1}{2}\pm \mathrm{i}\frac{\sqrt{3}}{2},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

… ►

4.4.12 $${\mathrm{i}}^{\pm \mathrm{i}}={\mathrm{e}}^{\mp \pi /2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit

Permalink:

http://dlmf.nist.gov/4.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.4(ii), §4.4 and Ch.4

…

25: 22.18 Mathematical Applications

… ►In polar coordinates, $x=r\cos \varphi$, $y=r\sin \varphi$, the lemniscate is given by $r^2=\cos \left(2\varphi \right)$, $0\le \varphi \le 2\pi$. The arc length $l(r)$, measured from $\varphi =0$, is ►

22.18.4 $$l(r)=(1/\sqrt{2})\mathrm{arccn}(r,1/\sqrt{2}).$$

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Symbols:

$\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

… ►

22.18.5 $$r=\mathrm{cn}(\sqrt{2}l,1/\sqrt{2}),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $r$: radius and $l(r)$: arc length

Permalink:

http://dlmf.nist.gov/22.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.18(i), §22.18(i), §22.18 and Ch.22

… ►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. …

26: 33.2 Definitions and Basic Properties

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§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

… ►

§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by … ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . Also, ${\mathrm{e}}^{\mp \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{H}_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are analytic functions of $\eta$ when . …

27: 33.13 Complex Variable and Parameters

… ►The functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, and $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ may be extended to noninteger values of $\mathrm{\ell }$ by generalizing $(2\mathrm{\ell }+1)!=\mathrm{\Gamma }\left(2\mathrm{\ell }+2\right)$, and supplementing (33.6.5) by a formula derived from (33.2.8) with $U(a,b,z)$ expanded via (13.2.42). … ►The quantities $C_{\mathrm{\ell }}\left(\eta \right)$, ${\sigma }_{\mathrm{\ell }}\left(\eta \right)$, and $R_{\mathrm{\ell }}$, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that … ►

33.13.2 $$R_{\mathrm{\ell }}=(2\mathrm{\ell }+1)C_{\mathrm{\ell }}\left(\eta \right)/C_{\mathrm{\ell }-1}\left(\eta \right).$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $R_{\mathrm{\ell }}$: factor

Permalink:

http://dlmf.nist.gov/33.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.13 and Ch.33

…

28: 22.19 Physical Applications

… ►The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta \in (0,\pm \alpha )$ and $\theta \in (\pm \alpha ,0)$ have the same “quarter periods” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►

►►

Figure 22.19.1: Jacobi’s amplitude function $\mathrm{am}(x,k)$ for $0\le x\le 10\pi$ and $k=0.5,0.9999,1.0001,2$. …As $k\to 1-$, plateaus are seen as the motion approaches the separatrix where $\theta =n\pi$, $n=\pm 1,\pm 2,\mathrm{\dots }$, at which points the motion is time independent for $k=1$. … Magnify

… ►As $a\to \sqrt{1/\beta }$ from below the period diverges since $a=\pm \sqrt{1/\beta }$ are points of unstable equilibrium. … ►For an initial displacement with , bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta }$. …For initial displacement with $|a|\ge \sqrt{2/\beta }$ the motion extends over the full range $-a\le x\le a$: …

29: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

30: 33.23 Methods of Computation

§33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. … ►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. …