functions f(ϵ,ℓ;r),h(ϵ,ℓ;r)

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41: 3.3 Interpolation

… ►Given $n+1$ distinct points $z_k$ and $n+1$ corresponding function values $f_k$, the Lagrange interpolation polynomial is the unique polynomial $P_n(z)$ of degree not exceeding $n$ such that $P_n(z_k)=f_k$, $k=0,1,\mathrm{\dots },n$. … ►where the nodes $x_k=x_0+kh$ ($h>0$) and function $f$ are real, … ►In this method we interchange the roles of the points $z_k$ and the function values $f_k$. … ►To compute the first negative zero $a_1=-\mathrm{2.33810\hspace{0.33em}7410}\mathrm{\dots }$ of the Airy function $f(x)=\mathrm{Ai}\left(x\right)$ (§9.2). … ►For interpolation of a bounded function $f$ on $ℝ$ the cardinal function of $f$ is defined by …

42: 15.6 Integral Representations

… ►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: ►

15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.6 $$𝐅(a,b;c;z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(c+t\right)}{(-z)}^{t}dt,$$ ; $a,b\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.3.2 (modified)

Referenced by:

§15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

… ►

15.6.8 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(c-d\right)}{\int }_0^1𝐅(a,b;d;zt)t^{d-1}{(1-t)}^{c-d-1}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }d>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Proof sketch:

Follows from (15.6.9) with $\lambda =a+b$.

Referenced by:

(15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)

Permalink:

http://dlmf.nist.gov/15.6.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

…

43: Software Index

… ► ►►►

	Open Source													With Book						Commercial
…
7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
…

…

44: 16.18 Special Cases

… ►The ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_p{}^{}F_q^{}$ functions introduced in the present chapter, are all special cases of the Meijer $G$-function. … ►

16.18.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\left(\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)/\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)\right)G_{p,q+1}^{1,p}(-z;\genfrac{}{}{0pt}{}{1-a_1,\mathrm{\dots },1-a_p}{0,1-b_1,\mathrm{\dots },1-b_q})=\left(\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)/\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)\right)G_{q+1,p}^{p,1}(-\frac{1}{z};\genfrac{}{}{0pt}{}{1,b_1,\mathrm{\dots },b_q}{a_1,\mathrm{\dots },a_p}).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.18 and Ch.16

►As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …

45: 27.3 Multiplicative Properties

… ►Except for $\nu \left(n\right)$, $\mathrm{\Lambda }\left(n\right)$, $p_n$, and $\pi \left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and ►

27.3.1 $$f(mn)=f(m)f(n),$$ $\left(m,n\right)=1$.

ⓘ

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer, $n$: positive integer and $f(n)$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►

27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

ⓘ

Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20, §27.3

Permalink:

http://dlmf.nist.gov/27.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

… ►A function $f$ is completely multiplicative if $f(1)=1$ and ►

27.3.9 $$f(mn)=f(m)f(n),$$ $m,n=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$m$: positive integer, $n$: positive integer and $f(n)$: multiplicative function

Permalink:

http://dlmf.nist.gov/27.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

…

46: 30.15 Signal Analysis

… ►

30.15.9 $$\beta =\frac{1}{2\pi }{\int }_{-\sigma }^{\sigma }{\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }f(t)dt\right|}^{2}d\omega$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sigma >0$: parameter and $f$: function

Permalink:

http://dlmf.nist.gov/30.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(v), §30.15 and Ch.30

… ►The corresponding function $f$ is given by …

47: 2.2 Transcendental Equations

… ►Let $f(x)$ be continuous and strictly increasing when and ►

2.2.1 $$f(x)\sim x,$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: asymptotic equality and $f(x)$: function

Referenced by:

§2.2

Permalink:

http://dlmf.nist.gov/2.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2 and Ch.2

… ►

2.2.7 $$f(x)\sim x+f_0+f_1{x}^{-1}+f_2{x}^{-2}+\mathrm{\cdots },$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function and $f_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

…

48: 4.7 Derivatives and Differential Equations

… ►For a nonvanishing analytic function $f(z)$, the general solution of the differential equation ►

4.7.5 $$\frac{dw}{dz}=\frac{f^{\prime }(z)}{f(z)}$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►

4.7.6 $$w(z)=\mathrm{Ln}\left(f(z)\right)+\text{ constant}.$$

ⓘ

Defines:

$w$: solution (locally)

Symbols:

$\mathrm{Ln}z$: general logarithm function, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►

4.7.12 $$\frac{dw}{dz}=f(z)w$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

… ►

4.7.13 $$w=\exp \left(\int f(z)dz\right)+\mathrm{constant}.$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

…

49: 2.1 Definitions and Elementary Properties

… ►For example, suppose $f(x)$ is continuous and $f(x)\sim x^{\nu }$ as $x\to +\mathrm{\infty }$ in $ℝ$, where $\nu$ ($\in ℂ$) is a constant. … ►Let $\sum a_s{x}^{-s}$ be a formal power series (convergent or divergent) and for each positive integer $n$, … ►But for any given set of coefficients $a_0,a_1,a_2,\mathrm{\dots }$, and suitably restricted $𝐗$ there is an infinity of analytic functions $f(x)$ such that (2.1.14) and (2.1.16) apply. … ►Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $𝐔$, and for each nonnegative integer $n$ … ►Suppose also that $f(x)$ and $f_s(x)$ satisfy …

50: 15.4 Special Cases

… ►

15.4.20 $$F(a,b;c;1)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.20

Referenced by:

§13.23(i), §15.4(ii), §15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.21 $$\underset{z\to 1-}{lim}\frac{F(a,b;a+b;z)}{-\ln \left(1-z\right)}=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\ln z$: principal branch of logarithm function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.4(ii)

Permalink:

http://dlmf.nist.gov/15.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.23 $$\underset{z\to 1-}{lim}\frac{F(a,b;c;z)}{{(1-z)}^{c-a-b}}=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.4(ii)

Permalink:

http://dlmf.nist.gov/15.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(ii), §15.4 and Ch.15

… ►

15.4.26 $$F(a,b;a-b+1;-1)=\frac{\mathrm{\Gamma }\left(a-b+1\right)\mathrm{\Gamma }\left(\frac{1}{2}a+1\right)}{\mathrm{\Gamma }\left(a+1\right)\mathrm{\Gamma }\left(\frac{1}{2}a-b+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $a$: real or complex parameter and $b$: real or complex parameter

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

… ►

15.4.31 $$F(a,\frac{1}{2}+a;\frac{3}{2}-2a;-\frac{1}{3})={\left(\frac{8}{9}\right)}^{-2a}\frac{\mathrm{\Gamma }\left(\frac{4}{3}\right)\mathrm{\Gamma }\left(\frac{3}{2}-2a\right)}{\mathrm{\Gamma }\left(\frac{3}{2}\right)\mathrm{\Gamma }\left(\frac{4}{3}-2a\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function and $a$: real or complex parameter

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

…