# special values of the variable

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##### 3: Bille C. Carlson
In his paper Lauricella’s hypergeometric function $F_{D}$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. If some of the parameters are equal, then the $R$-function is symmetric in the corresponding variables. …Also, the homogeneity of the $R$-function has led to a new type of mean value for several variables, accompanied by various inequalities. The foregoing matters are discussed in Carlson’s book Special Functions of Applied Mathematics, published by Academic Press in 1977. …
##### 4: Mathematical Introduction
With two real variables, special functions are depicted as 3D surfaces, with vertical height corresponding to the value of the function, and coloring added to emphasize the 3D nature. … Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …
##### 6: 35.1 Special Notation
###### §35.1 Special Notation
(For other notation see Notation for the Special Functions.) … All fractional or complex powers are principal values.
 $a,b$ complex variables. …
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively $\Gamma_{m}\left(a\right)$ and $\mathrm{B}_{m}\left(a,b\right)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu}\left(\mathbf{T}\right)$ and (of the second kind) $B_{\nu}\left(\mathbf{T}\right)$; confluent hypergeometric (of the first kind) ${{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)$ or $\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)$ and (of the second kind) $\Psi\left(a;b;\mathbf{T}\right)$; Gaussian hypergeometric ${{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)$ or $\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)$; generalized hypergeometric ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ or $\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)$. …
##### 7: 8.19 Generalized Exponential Integral
When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_{p}\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed. … In Figures 8.19.28.19.5, height corresponds to the absolute value of the function and color to the phase. …
##### 8: 4.31 Special Values and Limits
###### §4.31 SpecialValues and Limits
4.31.1 $\lim_{z\to 0}\frac{\sinh z}{z}=1,$
4.31.2 $\lim_{z\to 0}\frac{\tanh z}{z}=1,$
4.31.3 $\lim_{z\to 0}\frac{\cosh z-1}{z^{2}}=\frac{1}{2}.$
##### 9: 25.2 Definition and Expansions
25.2.2 $\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}},$ $\Re s>1$.
25.2.3 $\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^% {s}},$ $\Re s>0$.
##### 10: 25.11 Hurwitz Zeta Function
25.11.11 $\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right),$ $s\neq 1$.
25.11.15 $\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}% \right),$ $s\neq 1$, $k=1,2,3,\dots$.
25.11.16 $\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*% \sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta\left(s,% \frac{r}{k}\right),$ $s\neq 0,1$; $h,k$ integers, $1\leq h\leq k$.