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21: 28.1 Special Notation

§28.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The notation for the joining factors is … ►Alternative notations for the parameters $a$ and $q$ are shown in Table 28.1.1. … ►Alternative notations for the functions are as follows. …

22: 14.16 Zeros

… ►

§14.16(i) Notation

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23: 10.38 Derivatives with Respect to Order

§10.38 Derivatives with Respect to Order

… ►For the notations $E_1$ and $\mathrm{Ei}$ see §6.2(i). …

24: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.7 $$\frac{1}{\pi }{\int }_{{ℒ}_j}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu 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, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.18 (in slightly different notation)

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

… ►With the notations of §28.4 for $A_m^n(q)$ and $B_m^n(q)$, §28.14 for $c_n^{\nu }(q)$, and (28.23.1) for ${𝒞}_{\mu }^{(j)}$, $j=1,2,3,4$, … ►

§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

►With the parameter $h$ suppressed we use the notation … ►

§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

…

25: 13.7 Asymptotic Expansions for Large Argument

… ►and with the notation of Figure 13.7.1 … ►(For the notation see §8.2(i).) … ►

13.7.13

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $m$: integer, $n$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant and $R_n(a,b,z)$: remainder

Permalink:

http://dlmf.nist.gov/13.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(iii), §13.7 and Ch.13

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26: 13.8 Asymptotic Approximations for Large Parameters

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13.8.1 $$M(a,b,z)=\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s+O\left({|b|}^{-n}\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii)

Permalink:

http://dlmf.nist.gov/13.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(i), §13.8 and Ch.13

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13.8.6 $$M(a,b,b)=\sqrt{\pi }{\left(\frac{b}{2}\right)}^{\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(ii), §13.8 and Ch.13

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13.8.7 $$U(a,b,b)=\sqrt{\pi }{\left(2b\right)}^{-\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(ii), §13.8 and Ch.13

… ►For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv). … ►

13.8.17 $$M(a,b,z)={\mathrm{e}}^{\nu z}\frac{{\mathrm{\Gamma }}^{\ast }(b)}{{\mathrm{\Gamma }}^{\ast }(a)}\left(1+\frac{(1-\nu )(1+6{\nu }^2{z}^2)}{12a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iv), §13.8 and Ch.13

…

27: 9.7 Asymptotic Expansions

… ►

§9.7(i) Notation

… ►(For the notation see §8.2(i).) … ►

9.7.23 $$R_{m,n}(z),S_{m,n}(z)=O\left({\mathrm{e}}^{-2|\zeta |}{\zeta }^{-m}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable, $\zeta$: change of variable, $m$: index, $n$: index, $R_n$: remainder function and $S_n$: remainder function

Proof sketch:

Use the methods of Olver (1991b, 1993a).

Permalink:

http://dlmf.nist.gov/9.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(v), §9.7 and Ch.9

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28: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

►With real critical points (36.4.1) ordered so that … ►

36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

29: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$-symbol: … ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

30: 29.12 Definitions

… ►Throughout §§29.12–29.16 the order $\nu$ in the differential equation (29.2.1) is assumed to be a nonnegative integer. … ►There are eight types of Lamé polynomials, defined as follows: … ►

Table 29.12.1: Lamé polynomials.

► ►►

$\nu$	eigenvalue $h$	eigenfunction $w(z)$	polynomial form	real period	imag. period	parity of $w(z)$	parity of $w(z-K)$	parity of $w(z-K-\mathrm{i}{K}^{\prime })$
…

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