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21: 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 $-a\le x\le a$: …

22: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. …

23: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.1) the integration path may not pass through either of the points $t=\pm \mathrm{i}$, and the function ${(1+t^2)}^{1/2}$ assumes its principal value when $t$ is real. In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. …In (4.37.3) the integration path may not intersect $\pm 1$. …$\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►

4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

ⓘ

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

…

24: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.1 $$\mathrm{ci}(a,z)\pm \mathrm{i}\mathrm{si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }(a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}),$$

ⓘ

Defines:

$\mathrm{ci}(a,z)$: generalized cosine integral and $\mathrm{si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

►

8.21.2 $$\mathrm{Ci}(a,z)\pm \mathrm{i}\mathrm{Si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\gamma (a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}).$$

ⓘ

Defines:

$\mathrm{Ci}(a,z)$: generalized cosine integral and $\mathrm{Si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

… ►

8.21.3 $${\int }_0^{\mathrm{\infty }}{t}^{a-1}{\mathrm{e}}^{\pm \mathrm{i}t}dt={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\mathrm{\Gamma }\left(a\right),$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $a$: parameter

Referenced by:

§8.21(iv)

Permalink:

http://dlmf.nist.gov/8.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(ii), §8.21 and Ch.8

…

25: 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

►where $a=1+\mathrm{\ell }\pm \mathrm{i}\eta$ and $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§5.2(i)). … ►Corresponding expansions for $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).

26: 4.21 Identities

… ►

4.21.1 $$\sin u\pm \cos u=\sqrt{2}\sin \left(u\pm \frac{1}{4}\pi \right)=\pm \sqrt{2}\cos \left(u\mp \frac{1}{4}\pi \right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $\sin z$: sine function

Referenced by:

§4.21(i), Erratum (V1.0.7) for Equation (4.21.1)

Permalink:

http://dlmf.nist.gov/4.21.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.0.7):

Originally the symbol $\pm$ was missing after the second equal sign.

Suggested 2012-09-27 by Dennis M. Heim

See also:

Annotations for §4.21(i), §4.21 and Ch.4

… ►

4.21.2 $$\sin \left(u\pm v\right)=\sin u\cos v\pm \cos u\sin v,$$

ⓘ

Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.16

Referenced by:

§4.21(i), (9.8.13)

Permalink:

http://dlmf.nist.gov/4.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.3 $$\cos \left(u\pm v\right)=\cos u\cos v\mp \sin u\sin v,$$

ⓘ

Symbols:

$\cos z$: cosine function and $\sin z$: sine function

A&S Ref:

4.3.17

Referenced by:

§4.21(i)

Permalink:

http://dlmf.nist.gov/4.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.4 $$\tan \left(u\pm v\right)=\frac{\tan u\pm \tan v}{1\mp \tan u\tan v},$$

ⓘ

Symbols:

$\tan z$: tangent function

A&S Ref:

4.3.18

Referenced by:

(9.8.17)

Permalink:

http://dlmf.nist.gov/4.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

►

4.21.5 $$\cot \left(u\pm v\right)=\frac{\pm \cot u\cot v-1}{\cot u\pm \cot v}.$$

ⓘ

Symbols:

$\cot z$: cotangent function

A&S Ref:

4.3.19

Permalink:

http://dlmf.nist.gov/4.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.21(i), §4.21 and Ch.4

…

27: 32.7 Bäcklund Transformations

… ►and …furnish solutions of ${\text{P}}_{\text{II}}$, provided that $\alpha \ne \mp \frac{1}{2}$. … ►The solutions $w_{\alpha }=w(z;\alpha )$, $w_{\alpha \pm 1}=w(z;\alpha \pm 1)$, satisfy the nonlinear recurrence relation … ►Let $w_0=w(z;{\alpha }_0,{\beta }_0)$ and $w_j^{\pm }=w(z;{\alpha }_j^{\pm },{\beta }_j^{\pm })$, $j=1,2,3,4$, be solutions of ${\text{P}}_{\text{IV}}$ with … ►with $\epsilon =\pm 1$. …

28: 9.12 Scorer Functions

… ►

${\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)$ ,

… ►

9.12.10 $${\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{3}\pi$.

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Olver (1997b, p. 432)

Permalink:

http://dlmf.nist.gov/9.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iv), §9.12 and Ch.9

… ►

9.12.13 $$\mathrm{Gi}\left(z\right)={\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}\right)\pm \mathrm{i}\mathrm{Ai}\left(z\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Verify with the aid of §§9.2(ii) and 9.12(iii).

Permalink:

http://dlmf.nist.gov/9.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(v), §9.12 and Ch.9

►

9.12.14 $$\mathrm{Hi}\left(z\right)={\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}\right)+2{\mathrm{e}}^{\mp \pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right).$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Verify with the aid of §§9.2(ii) and 9.12(iii).

Referenced by:

§9.12(viii)

Permalink:

http://dlmf.nist.gov/9.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(v), §9.12 and Ch.9

… ►If $\zeta =\frac{2}{3}{z}^{3/2}$ or $\frac{2}{3}{x}^{3/2}$, and $K_{1/3}$ is the modified Bessel function (§10.25(ii)), then …

29: 10.19 Asymptotic Expansions for Large Order

… ►

10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

►with sectors of validity $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$. … ►

$P_2(a)=-\frac{9}{100}{a}^5+\frac{3}{35}{a}^2,$

… ►

10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

… ►

$R_2(a)=-\frac{9}{100}{a}^5+\frac{57}{70}{a}^2,$

…

30: 19.34 Mutual Inductance of Coaxial Circles

… ►

19.34.3 $$2abI({𝐞}_5)=a_{3}I(\mathrm{𝟎})-I({𝐞}_3)=a_{3}I(\mathrm{𝟎})-r_{+}^2{r}_{-}^{2}I(-{𝐞}_3)=2ab(I(\mathrm{𝟎})-r_{-}^{2}I({𝐞}_1-{𝐞}_3)),$$

ⓘ

Symbols:

$I(𝐦)$: integral and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

►where $a_1+b_{1}t=1+t$ and ►

19.34.4 $$r_{\pm }^2=a_3\pm 2ab=h^2+{(a\pm b)}^2$$

ⓘ

Symbols:

$h$: distance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►

19.34.5 $$\frac{3c^2}{8\pi ab}M=3R_F(0,r_{+}^2,r_{-}^2)-2r_{-}^2{R}_D(0,r_{+}^2,r_{-}^2),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Permalink:

http://dlmf.nist.gov/19.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

… ►

19.34.6 $$\frac{c^2}{2\pi }M=(r_{+}^2+r_{-}^2)R_F(0,r_{+}^2,r_{-}^2)-4R_G(0,r_{+}^2,r_{-}^2).$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $M$: mutual inductance and $r_{\pm }$

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.34 and Ch.19

…