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11: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … … ►On the circle of convergence, $|z|=1$, the Gauss series: … ►

§15.2(ii) Analytic Properties

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12: 5.12 Beta Function

§5.12 Beta Function

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Euler’s Beta Integral

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Figure 5.12.1: $t$-plane. Contour for first loop integral for the beta function. Magnify

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Figure 5.12.2: $t$-plane. Contour for second loop integral for the beta function. Magnify

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Pochhammer’s Integral

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13: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

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§14.20(i) Definitions and Wronskians

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§14.20(ii) Graphics

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§14.20(x) Zeros and Integrals

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14: 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).

15: 4.2 Definitions

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§4.2(iii) The Exponential Function

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§4.2(iv) Powers

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Powers with General Bases

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16: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ► $\mathrm{\wp }\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points. … … ►Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. …

17: 1.10 Functions of a Complex Variable

… ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. … … ►

§1.10(vi) Multivalued Functions

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§1.10(vii) Inverse Functions

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§1.10(xi) Generating Functions

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18: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

… ► $\zeta (s,a)$ has a meromorphic continuation in the $s$-plane, its only singularity in $ℂ$ being a simple pole at $s=1$ with residue $1$. …The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

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19: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

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§8.17(iv) Recurrence Relations

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§8.17(vi) Sums

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20: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
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primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.