Gauss theorem for vector-valued functions

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21: 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 $L(s,\chi )$.

22: 12.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …

23: 17.1 Special Notation

§17.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. ►Another function notation used is the “idem” function: …

24: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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§4.37(i) General Definitions

… ►Each of the six functions is a multivalued function of $z$. … ►

Other Inverse Functions

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§4.37(vi) Interrelations

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25: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

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§11.10(v) Interrelations

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§11.10(vi) Relations to Other Functions

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

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26: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(i) General Definitions

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Other Inverse Functions

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§4.23(viii) Gudermannian Function

… ►The inverse Gudermannian function is given by …

27: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

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§23.2(iii) Periodicity

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28: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(x) Modulus and Phase Functions

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29: 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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30: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). …