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21: 9.17 Methods of Computation

… ►However, in the case of $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►But when the integration has to be towards the origin, with starting values of $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ computed from their asymptotic expansions. … ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). ►Gil et al. (2002c) describes two methods for the computation of $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ for $z\in ℂ$. …The methods for ${\mathrm{Ai}}^{\prime }\left(z\right)$ are similar. …

22: 28.10 Integral Equations

… ►

28.10.3 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=-\frac{h{A}_1^{2n+1}(h^2)}{{\mathrm{ce}}_{2n+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

… ►

28.10.5 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+1}(t,h^2)dt=\frac{h{B}_1^{2n+1}(h^2)}{{\mathrm{se}}_{2n+1}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

… ►

28.10.7 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin z\sin t\sin \left(2h\cos z\cos t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=-\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{se}}_{2n+2}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.22 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.8 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+2}(z,h^2).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

…

23: 3.4 Differentiation

… ►and follows from the differentiated form of (3.3.4). … ►

3.4.5 $$h{f}_t^{\prime }=-f_0+f_1+h{R}_{1,t}^{\prime },$$ .

ⓘ

Symbols:

$R_{n,t}^{\prime }(x)$: remainder

Permalink:

http://dlmf.nist.gov/3.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►

3.4.7 $$h{f}_t^{\prime }=\sum _{k=-1}^2{B}_k^3{f}_k+h{R}_{3,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

A&S Ref:

25.3.5 (modification of)

Permalink:

http://dlmf.nist.gov/3.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►

3.4.11 $$h{f}_t^{\prime }=\sum _{k=-2}^3{B}_k^5{f}_k+h{R}_{5,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

Permalink:

http://dlmf.nist.gov/3.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

… ►

3.4.15 $$h{f}_t^{\prime }=\sum _{k=-3}^4{B}_k^7{f}_k+h{R}_{7,t}^{\prime },$$ ,

ⓘ

Symbols:

$B_k^n$: differentiated Lagrangian interpolation coefficients and $R_{n,t}^{\prime }(x)$: remainder

Permalink:

http://dlmf.nist.gov/3.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(i), §3.4(i), §3.4 and Ch.3

…

24: 22.15 Inverse Functions

… ►

22.15.3 $$\mathrm{dn}(\zeta ,k)=x,$$ $k^{\prime }\le x\le 1$,

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

… ►

§22.15(ii) Representations as Elliptic Integrals

… ►The integrals (22.15.12)–(22.15.14) can be regarded as normal forms for representing the inverse functions. …can be transformed into normal form by elementary change of variables. … ►

25: 9.2 Differential Equation

… ►

9.2.6 $${\mathrm{Bi}}^{\prime }\left(0\right)=\frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}=\mathrm{0.44828\hspace{0.33em}83573}\mathrm{\dots }.$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.5 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

…

26: 9.6 Relations to Other Functions

… ►

9.6.12 $$J_{\pm 2/3}\left(\zeta \right)=\frac{1}{2}(\sqrt{3}/z)\left(\pm \sqrt{3}{\mathrm{Ai}}^{\prime }\left(-z\right)+{\mathrm{Bi}}^{\prime }\left(-z\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.27

Permalink:

http://dlmf.nist.gov/9.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

… ►

9.6.14 $$I_{\pm 2/3}\left(\zeta \right)=\frac{1}{2}(\sqrt{3}/z)\left(\pm \sqrt{3}{\mathrm{Ai}}^{\prime }\left(z\right)+{\mathrm{Bi}}^{\prime }\left(z\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.30

Permalink:

http://dlmf.nist.gov/9.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

… ►

9.6.16 $$K_{\pm 2/3}\left(\zeta \right)=-\pi (\sqrt{3}/z){\mathrm{Ai}}^{\prime }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.31

Permalink:

http://dlmf.nist.gov/9.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

►

9.6.17 $$H_{1/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\sqrt{3/z}\left(\mathrm{Ai}\left(-z\right)-\mathrm{i}\mathrm{Bi}\left(-z\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.23 (with different sign, different form!)

Referenced by:

(9.8.11), (9.8.9)

Permalink:

http://dlmf.nist.gov/9.6.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

►

9.6.18 $$H_{2/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{-2/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{\pi \mathrm{i}/6}(\sqrt{3}/z)\left({\mathrm{Ai}}^{\prime }\left(-z\right)-\mathrm{i}{\mathrm{Bi}}^{\prime }\left(-z\right)\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.28

Referenced by:

(9.8.10), (9.8.12)

Permalink:

http://dlmf.nist.gov/9.6.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

…

27: 3.7 Ordinary Differential Equations

… ►By repeated differentiation of (3.7.1) all derivatives of $w(z)$ can be expressed in terms of $w(z)$ and $w^{\prime }(z)$ as follows. … ►The remaining two equations are supplied by boundary conditions of the form … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. … ►For $w^{\prime }=f(z,w)$ the standard fourth-order rule reads … ►For $w^{\prime \prime }=f(z,w,w^{\prime })$ the standard fourth-order rule reads …

28: 3.3 Interpolation

… ►Here the prime signifies that the factor for $j=k$ is to be omitted, ${\delta }_{k,j}$ is the Kronecker symbol, and ${\omega }_{n+1}(z)$ is the nodal polynomial …The final expression in (3.3.1) is the Barycentric form of the Lagrange interpolation formula. … … ►The $(n+1)$-point formula (3.3.4) can be written in the form … ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. …

29: 22.5 Special Values

… ►Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $\mathrm{i}{K}^{\prime }$. … ►For example, $\mathrm{sn}(\frac{1}{2}K,k)={(1+k^{\prime })}^{-1/2}$. … ►

§22.5(ii) Limiting Values of $k$

… ►Expansions for $K,K^{\prime }$ as $k\to 0$ or $1$ are given in §§19.5, 19.12. … ►

30: 25.11 Hurwitz Zeta Function

… ►

25.11.15 $$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form.

Referenced by:

(25.11.24)

Permalink:

http://dlmf.nist.gov/25.11.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

… ►In (25.11.18)–(25.11.24) primes on $\zeta$ denote derivatives with respect to $s$. … ►

25.11.34 $$n{\int }_0^a{\zeta }^{\prime }(1-n,x)dx={\zeta }^{\prime }(-n,a)-{\zeta }^{\prime }\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $a$: real or complex parameter

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (15), p. 196)

Permalink:

http://dlmf.nist.gov/25.11.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

… ►For the more general case ${\zeta }^{\prime }(-m,a)$, $m=1,2,\mathrm{\dots }$, see Elizalde (1986). ►For an exponentially-improved form of (25.11.43) see Paris (2005b).