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31: 9.16 Physical Applications

… ►Airy functions are applied in many branches of both classical and quantum physics. … ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. … ►Other applications appear in the study of instability of Couette flow of an inviscid fluid. … ►The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. … ►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

32: 2.3 Integrals of a Real Variable

… ►

2.3.3 $${\sigma }_n=\underset{(0,\mathrm{\infty })}{sup}(t^{-1}\ln |q^{(n)}(t)/q^{(n)}(0)|)$$

ⓘ

Defines:

${\sigma }_n$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.3.E3

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Annotations for §2.3(i), §2.3 and Ch.2

… ►For other examples, see Wong (1989, Chapter 1). … ►Other types of singular behavior in the integrand can be treated in an analogous manner. …(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) … ►For other estimates of the error term see Lyness (1971). …

33: 22.14 Integrals

… ►

22.14.1 $$\int \mathrm{sn}(x,k)dx=k^{-1}\ln \left(\mathrm{dn}(x,k)-k\mathrm{cn}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.1

Permalink:

http://dlmf.nist.gov/22.14.E1

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Annotations for §22.14(i), §22.14 and Ch.22

… ►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►

22.14.4 $$\int \mathrm{cd}(x,k)dx=k^{-1}\ln \left(\mathrm{nd}(x,k)+k\mathrm{sd}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.4

Permalink:

http://dlmf.nist.gov/22.14.E4

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Annotations for §22.14(i), §22.14 and Ch.22

… ►Again, the branches of the inverse trigonometric functions must be continuous. … ►

§22.14(iii) Other Indefinite Integrals

…

34: 10.17 Asymptotic Expansions for Large Argument

… ►where the branch of $z^{\frac{1}{2}}$ is determined by ►

10.17.7 $$z^{\frac{1}{2}}=\exp \left(\frac{1}{2}\ln |z|+\frac{1}{2}\mathrm{i}\mathrm{ph}z\right).$$

ⓘ

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.17.E7

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Annotations for §10.17(i), §10.17 and Ch.10

►Corresponding expansions for other ranges of $\mathrm{ph}z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). …

35: 3.10 Continued Fractions

… ►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5). For example, by converting the Maclaurin expansion of $\arctan z$ (4.24.3), we obtain a continued fraction with the same region of convergence ($\left|z\right|\le 1$, $z\ne \pm \mathrm{i}$), whereas the continued fraction (4.25.4) converges for all $z\in ℂ$ except on the branch cuts from $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$ and $-\mathrm{i}$ to $-\mathrm{i}\mathrm{\infty }$. …

36: 24.16 Generalizations

… ►

24.16.4 $${\left(\frac{\ln \left(1+t\right)}{t}\right)}^{\mathrm{\ell }}=\mathrm{\ell }\sum _{n=0}^{\mathrm{\infty }}\frac{B_n^{(\mathrm{\ell }+n)}}{\mathrm{\ell }+n}\frac{t^n}{n!},$$ .

ⓘ

Symbols:

$!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: integer, $n$: integer and $t$: real or complex

Referenced by:

§24.16(i)

Permalink:

http://dlmf.nist.gov/24.16.E4

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Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

►For this and other properties see Milne-Thomson (1933, pp. 126–153) or Nörlund (1924, pp. 144–162). … ►

24.16.5 $$\frac{t}{\ln \left(1+t\right)}=\sum _{n=0}^{\mathrm{\infty }}{b}_n{t}^n,$$ ,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $n$: integer, $t$: real or complex and $b_n$

Permalink:

http://dlmf.nist.gov/24.16.E5

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Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

… ►

§24.16(iii) Other Generalizations

►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

37: 32.9 Other Elementary Solutions

§32.9 Other Elementary Solutions

… ►

32.9.2 $$w(z;0,-2\kappa ,0,4\kappa \mu -{\lambda }^2)=z(\kappa {(\ln z)}^2+\lambda \ln z+\mu ),$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $z$: real, $\kappa$: constant, $\lambda$: constant and $\mu$: constant

Permalink:

http://dlmf.nist.gov/32.9.E2

Encodings:

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Annotations for §32.9(i), §32.9 and Ch.32

…

38: 4.9 Continued Fractions

… ►

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

ⓘ

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

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

►

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

ⓘ

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

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

… ►For other continued fractions involving logarithms see Lorentzen and Waadeland (1992, pp. 566–568). … ►For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). …

39: 19.9 Inequalities

… ►

19.9.2

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E2

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Annotations for §19.9(i), §19.9 and Ch.19

►

19.9.3

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E3

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Annotations for §19.9(i), §19.9 and Ch.19

… ►

19.9.5

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E5

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Annotations for §19.9(i), §19.9 and Ch.19

… ►Other inequalities are: … ►Other inequalities for $F(\varphi ,k)$ can be obtained from inequalities for $R_F(x,y,z)$ given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

40: 18.38 Mathematical Applications

… ►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … ►The Askey–Gasper inequality … ►

§18.38(iii) Other OP’s

… ► … ► …