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11: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

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§14.19(ii) Hypergeometric Representations

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§14.19(iv) Sums

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§14.19(v) Whipple’s Formula for Toroidal Functions

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12: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

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§15.2(ii) Analytic Properties

… ►The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. … ►For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial: …

13: 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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14: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative 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: 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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17: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

… ►The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

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§25.11(vi) Derivatives

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18: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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§12.14(iii) Graphs

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Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(viii) Asymptotic Expansions for Large Variable

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19: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►The associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$ is defined by … ►

§11.10(vi) Relations to Other Functions

… ►where … ►For $n=1,2,3,\mathrm{\dots }$, …

20: 1.10 Functions of a Complex Variable

§1.10 Functions of a Complex Variable

… ►Equalities hold iff $f(z)=Az$, where $A$ is a constant such that $\left|A\right|=M/R$. ►

§1.10(vi) Multivalued Functions

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§1.10(xi) Generating Functions

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