general%0Aelliptic functions

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21: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

22: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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§4.37(i) General Definitions

►The general values of the inverse hyperbolic functions are defined by … ►the upper or lower sign being taken according as $\mathrm{\Re }z\gtrless 0$; compare Figure 4.37.1(ii). … ►With $k\in \mathbb{Z}$, the general solutions of the equations …

23: 23.2 Definitions and Periodic Properties

… ►If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators. ►The generators of a given lattice $𝕃$ are not unique. …In general, if … ►

§23.2(ii) Weierstrass Elliptic Functions

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24: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(i) General Definitions

►The general values of the inverse trigonometric functions are defined by … ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. … ►With $k\in \mathbb{Z}$, the general solutions of the equations …

25: 1.10 Functions of a Complex Variable

… ►Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). … ►An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\ge 1$) at $z_0$ if the first nonzero coefficient in its Taylor series at $z_0$ is that of ${(z-z_0)}^m$. … ►Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ►(Or more generally, a simple contour that starts at the center and terminates on the boundary.) … ►

§1.10(xi) Generating Functions

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26: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►In other cases the general theory of (12.2.2) is available. … ►

§12.14(ii) Values at $z=0$ and Wronskian

… ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►When $x>0$ …

27: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►As a function of $a$, with $s$ ($\ne 1$) fixed, $\zeta (s,a)$ is analytic in the half-plane $\mathrm{\Re }a>0$. … ►Most references treat real $a$ with . … ►Throughout this subsection $\mathrm{\Re }a>0$. … ►For the more general case ${\zeta }^{\prime }(-m,a)$, $m=1,2,\mathrm{\dots }$, see Elizalde (1986). …

28: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►(11.10.4) also applies when $\mathrm{\Re }z=0$ and $\mathrm{\Re }\nu >0$. … ►

§11.10(vi) Relations to Other Functions

… ► … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

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29: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. … ►

§8.17(ii) Hypergeometric Representations

… ►With $a>0$, $b>0$, and , … ►

§8.17(vi) Sums

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30: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

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