special values of the variable

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1: 22.5 Special Values

§22.5 Special Values

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Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

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	$z$
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Table 22.5.2: Other special values of Jacobian elliptic functions.

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	$z$
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2: 22.8 Addition Theorems

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§22.8(iii) Special Relations Between Arguments

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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. …

5: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6 Symmetry, Special Values, and Limits to Monomials

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§18.6(i) Symmetry and Special Values

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Laguerre

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Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
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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.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, 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) ${}_{1}F_1(a;b;\mathbf{T})$ or ${}_{1}F_1({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};\mathbf{T})$ and (of the second kind) $\mathrm{\Psi }(a;b;\mathbf{T})$; Gaussian hypergeometric ${}_{2}F_1(a_1,a_2;b;\mathbf{T})$ or ${}_{2}F_1({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};\mathbf{T})$; generalized hypergeometric ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;\mathbf{T})$ or ${}_{p}F_q({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};\mathbf{T})$. …

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 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 8.19.2: $E_{\frac{1}{2}}\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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Figure 8.19.3: $E_1\left(x+\mathrm{i}y\right)$, $-4\le x\le 4$, $-4\le y\le 4$. Principal value. … Magnify 3D Help

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§8.19(iii) Special Values

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8: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
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4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

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Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

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4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

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Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

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4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

9: 25.2 Definition and Expansions

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25.2.2 $$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Derivable from (25.2.1).

A&S Ref:

23.2.20 (is the special case with integer values of $s$)

Referenced by:

(25.11.11)

Permalink:

http://dlmf.nist.gov/25.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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25.2.3 $$\zeta \left(s\right)=\frac{1}{1-2^{1-s}}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n^s},$$ $\mathrm{\Re }s>0$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Apostol (1976, (27), p. 292)

A&S Ref:

23.2.19 (is the special case with integer values of $s$)

Referenced by:

§25.11(x)

Permalink:

http://dlmf.nist.gov/25.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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10: 25.11 Hurwitz Zeta Function

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25.11.11 $$\zeta (s,\frac{1}{2})=(2^s-1)\zeta \left(s\right),$$ $s\ne 1$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

specialvalue

Source:

Derivable using (25.11.1), (25.2.2).

Referenced by:

§25.11(iv)

Permalink:

http://dlmf.nist.gov/25.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.15 $$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

specialvalue

Source:

Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form.

Referenced by:

(25.11.24)

Permalink:

http://dlmf.nist.gov/25.11.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.16 $$\zeta (1-s,\frac{h}{k})=\frac{2\mathrm{\Gamma }\left(s\right)}{{(2\pi k)}^s}\sum _{r=1}^k\cos \left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta (s,\frac{r}{k}),$$ $s\ne 0,1$; $h,k$ integers, $1\le h\le k$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $k$: nonnegative integer and $s$: complex variable

Keywords:

specialvalue

Source:

Apostol (1976, (14), p. 261)

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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