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31: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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Figure 28.5.2: ${\mathrm{fe}}_0(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_0(x,1)$. Magnify

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Figure 28.5.3: ${\mathrm{fe}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_1(x,0.5)$. Magnify

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Figure 28.5.4: ${\mathrm{fe}}_1(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{ce}}_1(x,1)$. Magnify

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Figure 28.5.5: ${\mathrm{ge}}_1(x,0.5)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{se}}_1(x,0.5)$. Magnify

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Figure 28.5.6: ${\mathrm{ge}}_1(x,1)$ for $0\le x\le 2\pi$ and (for comparison) ${\mathrm{se}}_1(x,1)$. Magnify

32: 18.5 Explicit Representations

… ►With $x=\cos \theta =\frac{1}{2}(z+z^{-1})$, … ►

Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

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$p_n(x)$	$w(x)$	$F(x)$	${\kappa }_n$
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$P_n\left(x\right)$	$1$	$1-x^2$	${(-2)}^{n}n!$
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… ►The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $P_n^{(\alpha ,\beta )}\left(x\right)$ when the conditions $\alpha >-1$ and $\beta >-1$ are not satisfied. …Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. … ►

$L_1\left(x\right)=-x+1,$

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33: 14.7 Integer Degree and Order

… ►where $W_{-1}(x)=0$, and for $n\ge 1$, … ►When and , … ►When and $|h|>1$, … ►When $x>1$, (14.7.19) applies with . … ►Lastly, when $x>1$, (14.7.21) applies with $|h|>x+{\left(x^2-1\right)}^{1/2}$. …

34: 8.17 Incomplete Beta Functions

… ►Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. … ►With $a>0$, $b>0$, and , …where and the branches of $s^{-a}$ and ${(1-s)}^{-b}$ are continuous on the path and assume their principal values when $s=c$. … ►The expansion (8.17.22) converges rapidly for . For $x>(a+1)/(a+b+2)$ or , more rapid convergence is obtained by computing $I_{1-x}(b,a)$ and using (8.17.4). …

35: 15.3 Graphics

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Figure 15.3.1: $F(\frac{4}{3},\frac{9}{16};\frac{14}{5};x),-100\le x\le 1$. Magnify

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

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Figure 15.3.4: $F(5,10;1;x),-1\le x\le 0.022$. Magnify

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36: 8.10 Inequalities

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8.10.1 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)\le 1,$$ $x>0$, ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable and $a$: parameter

Referenced by:

§8.10

Permalink:

http://dlmf.nist.gov/8.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

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8.10.3 $$x^{1-a}{\mathrm{e}}^x\mathrm{\Gamma }(a,x)=1+\frac{a-1}{x}\vartheta ,$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $\vartheta$

Permalink:

http://dlmf.nist.gov/8.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.10 and Ch.8

►then $\vartheta \to 1$ as $x\to \mathrm{\infty }$, and … ►

$A_1={\displaystyle \frac{x}{x+1-a}},$

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$B_1={\displaystyle \frac{x+1}{x+2-a}},$

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37: 14.14 Continued Fractions

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14.14.1 $$\frac{1}{2}{\left(x^2-1\right)}^{1/2}\frac{P_{\nu }^{\mu }\left(x\right)}{P_{\nu }^{\mu -1}\left(x\right)}=\frac{x_0}{y_0+}\frac{x_1}{y_1+}\frac{x_2}{y_2+}\mathrm{\cdots },$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

… ►provided that $x_{k+1}$ and $y_k$ do not vanish simultaneously for any $k=0,1,2,\mathrm{\dots }$. ►

14.14.3 $$(\nu -\mu )\frac{Q_{\nu }^{\mu }\left(x\right)}{Q_{\nu -1}^{\mu }\left(x\right)}=\frac{x_0}{y_0-}\frac{x_1}{y_1-}\frac{x_2}{y_2-}\mathrm{\cdots },$$ $\nu \ne \mu$,

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Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $x$: real variable, $\mu$: general order, $\nu$: general degree, $x_k$: numerator and $x_k$: denominator

Referenced by:

§14.14

Permalink:

http://dlmf.nist.gov/14.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.14 and Ch.14

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$y_k=(2\nu +2k+1)x,$

►again provided $x_{k+1}$ and $y_k$ do not vanish simultaneously for any $k=0,1,2,\mathrm{\dots }$.

38: 36.1 Special Notation

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$l,m,n$	integers.
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$𝐱$	$\{x_1,x_2,\mathrm{\dots },x_K\}$, where $x_1,x_2,\mathrm{\dots },x_K$ are real parameters; also $x_1=x$, $x_2=y$, $x_3=z$ when $K\le 3$.
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39: 25.13 Periodic Zeta Function

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25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

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See also:

Annotations for §25.13 and Ch.25

►where $\mathrm{\Re }s>1$ if $x$ is an integer, $\mathrm{\Re }s>0$ otherwise. ► $F(x,s)$ is periodic in $x$ with period 1, and equals $\zeta \left(s\right)$ when $x$ is an integer. … ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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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, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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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, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

40: 4.29 Graphics

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. ($\mathrm{arccosh}x$ is complex when .) Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. ($\mathrm{arctanh}x$ is complex when or $x>1$, and $\mathrm{arccoth}x$ is complex when .) Magnify

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. ($\mathrm{arcsech}x$ is complex when and $x>1$.) Magnify

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