DLMF: Untitled Document

… ►

31.10.2

\rho(t)=t^{\gamma-1}(t-1)^{\delta-1}(t-a)^{\epsilon-1},

ⓘ

Defines:: $\rho(t)$ : weight function (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Permalink:: http://dlmf.nist.gov/31.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10 and Ch.31

… ►

31.10.9

\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\[1.0pt] 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos}^{2}\theta\\[1.0pt] 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\[1.0pt] 0&0&-\frac{1}{2}+\delta+\sigma&{\cos}^{2}\phi\\[1.0pt] 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},

… ►

31.10.13

\rho(s,t)=(s-t)(st)^{\gamma-1}\left((1-s)(1-t)\right)^{\delta-1}\*\left((1-(s/% a))(1-(t/a))\right)^{\epsilon-1},

ⓘ

Defines:: $\rho(s,t)$ : weight function (locally)
Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter and $a$ : complex parameter
Permalink:: http://dlmf.nist.gov/31.10.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(ii), §31.10 and Ch.31

… ►

31.10.19

\mathcal{K}(u,v,w)=u^{1-\gamma}v^{1-\delta}w^{1-\epsilon}\mathscr{C}_{1-\gamma% }\left(u\sqrt{\sigma_{1}}\right)\*\mathscr{C}_{1-\delta}\left(v\sqrt{\sigma_{2% }}\right)\mathscr{C}_{1-\epsilon}\left(\mathrm{i}w\sqrt{\sigma_{1}+\sigma_{2}}% \right),

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $\mathrm{i}$ : imaginary unit, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\mathcal{K}(z;s,t)$ : kernel, $u$ , $v$ , $w$ and $\sigma_{1}$ , $\sigma_{2}$ : separation constants
Permalink:: http://dlmf.nist.gov/31.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

… ►

31.10.22

\mathcal{K}(r,\theta,\phi)=r^{m}{\sin}^{2p}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos}^{2}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos}^{2}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},

…

… ►

28.28.20

\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell}(2hR)\cos\left(2\ell\phi% \right)\operatorname{ce}_{2m}\left(t,h^{2}\right)\,\mathrm{d}t=\varepsilon_{% \ell}(-1)^{\ell+m}A^{2m}_{2\ell}(h^{2}){\operatorname{Mc}^{(j)}_{2m}}\left(z,h% \right),

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions, $\phi(z,t)$ : function, $R(z,t)$ : function, $\varepsilon_{\ell}$ : constants and $A_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.7.26 (in different form and only for $\ell=0$ )
Permalink:: http://dlmf.nist.gov/28.28.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(ii), §28.28 and Ch.28

…

… ►

2.5.15

I(x)=-\sum_{s=0}^{2n}(a_{s}\ln x+b_{s})x^{-s}+O\left(x^{-2n-1+\epsilon}\right),

n=0,1,2,\dots

.

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $a_{n}$ : coefficients, $b_{n}$ : coefficients, $\epsilon$ : parameter and $I(x)$ : convolution integral
Permalink:: http://dlmf.nist.gov/2.5.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

… ►

2.5.16

I(x)=\sum_{s=0}^{n-1}(c_{s}\ln x+d_{s})x^{-2s-1}+O\left(x^{-2n-1+\epsilon}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $\epsilon$ : parameter, $c$ : point, $I(x)$ : convolution integral and $d$ : point
Permalink:: http://dlmf.nist.gov/2.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

…

… ►

32.10.4

w(z;\tfrac{1}{2}\varepsilon)=-\varepsilon\phi^{\prime}(z)/\phi(z),

ⓘ

Symbols:: $z$ : real, $\varepsilon=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(ii), §32.10 and Ch.32

… ►

32.10.13

w(z)=-\varepsilon_{1}\phi^{\prime}(z)/\phi(z),

ⓘ

Symbols:: $z$ : real, $\varepsilon_{j}=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iii), §32.10 and Ch.32

… ►

32.10.18

w(z)=-\varepsilon\phi^{\prime}(z)/\phi(z),

ⓘ

Symbols:: $z$ : real, $\varepsilon=\pm 1$ and $\phi(z)$ : function
Permalink:: http://dlmf.nist.gov/32.10.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iv), §32.10 and Ch.32

… ►

32.10.19

\phi(z)=\left(C_{1}U\left(a,\sqrt{2}z\right)+C_{2}V\left(a,\sqrt{2}z\right)% \right)\exp\left(\tfrac{1}{2}\varepsilon z^{2}\right),

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : real, $\varepsilon=\pm 1$ , $\phi(z)$ : function, $a$ and $C$ : constant
Permalink:: http://dlmf.nist.gov/32.10.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iv), §32.10 and Ch.32

… ►

32.10.22

w(z)=\begin{cases}\dfrac{2\exp\left(z^{2}\right)}{\sqrt{\pi}\left(C-i% \operatorname{erfc}\left(iz\right)\right)},&\varepsilon=1,\\ \dfrac{2\exp\left(-z^{2}\right)}{\sqrt{\pi}\left(C-\operatorname{erfc}\left(z% \right)\right)},&\varepsilon=-1,\end{cases}

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $z$ : real, $\varepsilon=\pm 1$ and $C$ : constant
Permalink:: http://dlmf.nist.gov/32.10.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §32.10(iv), §32.10 and Ch.32

…

… ►

2.7.21

w_{1}(x)=f^{-1/4}(x)\exp\left(\int f^{1/2}(x)\,\mathrm{d}x\right)\*\left(1+% \epsilon_{1}(x)\right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\epsilon_{j}(x)$ : function and $w_{j}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/2.7.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

►

2.7.22

w_{2}(x)=f^{-1/4}(x)\exp\left(-\int f^{1/2}(x)\,\mathrm{d}x\right)\*\left(1+% \epsilon_{2}(x)\right),

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\epsilon_{j}(x)$ : function and $w_{j}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/2.7.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

►such that ►

2.7.23

|\epsilon_{j}(x)|,\;\;\tfrac{1}{2}f^{-1/2}(x)|\epsilon_{j}^{\prime}(x)|\leq% \exp\left(\tfrac{1}{2}\mathcal{V}_{a_{j},x}\left(F\right)\right)-1,

j=1,2

,

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, $(a_{1},a_{2})$ : interval, $\epsilon_{j}(x)$ : function and $F$ : error-control function
Permalink:: http://dlmf.nist.gov/2.7.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

…

… ►

13.7.4

U\left(a,b,z\right)=z^{-a}\sum_{s=0}^{n-1}\frac{{\left(a\right)_{s}}{\left(a-b% +1\right)_{s}}}{s!}(-z)^{-s}+\varepsilon_{n}(z),

ⓘ

Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer, $z$ : complex variable and $\varepsilon_{n}(z)$ : function
Referenced by:: §13.2(i), §13.7(ii)
Permalink:: http://dlmf.nist.gov/13.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §13.7(ii), §13.7 and Ch.13

… ►

13.7.5

\left|\varepsilon_{n}(z)\right|,~{}\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)% \right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{% n}}}{n!z^{a+n}}\right|\exp\left(\frac{2\alpha\rho C_{1}}{|z|}\right),

ⓘ

Defines:: $\varepsilon_{n}(z)$ : function (locally)
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\exp\NVar{z}$ : exponential function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $z$ : complex variable, $C_{n}$ : coefficient, $\alpha$ , $\beta$ and $\rho$
Referenced by:: §13.2(i)
Permalink:: http://dlmf.nist.gov/13.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §13.7(ii), §13.7 and Ch.13

…

… ►The

\epsilon_{n}

are the observable energies of the system, and an increasing function of

n

. … ►

18.39.10

\Psi(x,t)=\exp\left(\ifrac{-\mathrm{i}\epsilon_{n}t}{\hbar}\right)\psi_{n}(x),

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $(\NVar{a},\NVar{b})$ : open interval, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.39.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

… ►

18.39.12

\Psi(x,t)=\sum_{n=0}^{\infty}c_{n}\exp\left(\ifrac{-\mathrm{i}\epsilon_{n}t}{% \hbar}\right)\psi_{n}(x),

ⓘ

Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $(\NVar{a},\NVar{b})$ : open interval, $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.39.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

… ►With

N\to\infty

the functions normalized as

\delta(\epsilon-\epsilon^{\prime})

with measure

\,\mathrm{d}r

are, formally, …

… ►

1.17.15

\delta\left(x-a\right)=\int_{0}^{\infty}s\left(x,\ell;r\right)s\left(a,\ell;r% \right)\,\mathrm{d}r,

a>0

,

x>0

.

ⓘ

Symbols:: $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.17.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

…

… ►

31.3.5

z^{1-\gamma}\mathit{H\!\ell}\left(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha+1-% \gamma,\beta+1-\gamma,2-\gamma,\delta;z\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.3(iii), §31.7(ii)
Permalink:: http://dlmf.nist.gov/31.3.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(i), §31.3 and Ch.31

… ►

31.3.7

(1-z)^{1-\delta}\*\mathit{H\!\ell}\left(1-a,((1-a)\gamma+\epsilon)(1-\delta)+% \alpha\beta-q;\alpha+1-\delta,\beta+1-\delta,2-\delta,\gamma;1-z\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

… ►

31.3.8

\mathit{H\!\ell}\left(\frac{a}{a-1},\frac{\alpha\beta a-q}{a-1};\alpha,\beta,% \epsilon,\delta;\frac{a-z}{a-1}\right),

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

►

31.3.9

\left(\frac{a-z}{a-1}\right)^{1-\epsilon}\mathit{H\!\ell}\left(\frac{a}{a-1},% \frac{(a(\delta+\gamma)-\gamma)(1-\epsilon)}{a-1}+\frac{\alpha\beta a-q}{a-1};% \alpha+1-\epsilon,\beta+1-\epsilon,2-\epsilon,\delta;\frac{a-z}{a-1}\right).

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/31.3.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

… ►

31.3.10

z^{-\alpha}\mathit{H\!\ell}\left(\frac{1}{a},\frac{q}{a}-\alpha\left(\beta-% \epsilon\right)-\frac{\alpha}{a}\left(\beta-\delta\right);\alpha,\alpha-\gamma% +1,\alpha-\beta+1,\delta;\frac{1}{z}\right),

ⓘ

Symbols:: $\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{\alpha},\NVar{\beta},\NVar{% \gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter and $\beta$ : real or complex parameter
Referenced by:: §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.3.E10
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: The second entry in the $\mathit{H\!\ell}$ has been corrected with an extra minus sign.
See also:: Annotations for §31.3(ii), §31.3 and Ch.31

…

… ►

L. N. Nosova and S. A. Tumarkin (1965) Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations

\epsilon(py^{\prime})^{\prime}+(q+\epsilon r)y=f

. Pergamon Press, Oxford.

ⓘ

Notes:: D. E. Brown, translator.
External Links:: MathReview, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

…

epsilon function

21—30 of 63 matching pages

21: 31.10 Integral Equations and Representations

22: 28.28 Integrals, Integral Representations, and Integral Equations

23: 2.5 Mellin Transform Methods

24: 32.10 Special Function Solutions

25: 2.7 Differential Equations

26: 13.7 Asymptotic Expansions for Large Argument

27: 18.39 Applications in the Physical Sciences

28: 1.17 Integral and Series Representations of the Dirac Delta

29: 31.3 Basic Solutions

30: Bibliography N