Gauss theorem for vector-valued functions

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11: 11.9 Lommel Functions

§11.9 Lommel Functions

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Reflection Formulas

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§11.9(ii) Expansions in Series of Bessel Functions

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12: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iii) Translation of the Argument by Half-Periods

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§20.2(iv) $z$-Zeros

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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: 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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15: 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).

16: 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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17: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ►Equivalently, the function is denoted by ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$ or ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$, and sometimes, for brevity, by ${}_p{}^{}F_q^{}\left(z\right)$. … ► … ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). … ►See §16.5 for the definition of ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ as a contour integral when $p>q+1$ and none of the $a_k$ is a nonpositive integer. …

18: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

… ►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). ►

§8.17(iii) Integral Representation

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§8.17(vi) Sums

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19: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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§1.10(xi) Generating Functions

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20: 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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