locally analytic

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1: 32.2 Differential Equations

… ►be a nonlinear second-order differential equation in which $F$ is a rational function of $w$ and $dw/dz$, and is locally analytic in $z$, that is, analytic except for isolated singularities in $ℂ$. … ►in which $a(z)$, $b(z)$, $c(z)$, $d(z)$, and $\varphi (z)$ are locally analytic functions. …

2: 1.13 Differential Equations

… ►

1.13.15 $$H(z)=\frac{1}{4}{f}^2(z)+\frac{1}{2}{f}^{\prime }(z)-g(z).$$

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Defines:

$H(z)$ (locally)

Symbols:

$z$: variable, $f(z)$: analytic coefficient and $g(z)$: analytic coefficient

Permalink:

http://dlmf.nist.gov/1.13.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

1.13.16 $$\eta =\int \exp \left(-\int f(z)dz\right)dz.$$

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Defines:

$\eta$: change of variable (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/1.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

…

3: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . …

4: 2.5 Mellin Transform Methods

… ►

2.5.1 $$ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt,$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $dx$: differential of $x$ , $\int$: integral and $f(x)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E1

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

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5: 4.7 Derivatives and Differential Equations

… ►

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

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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

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6: 31.3 Basic Solutions

… ►If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion …

7: 28.11 Expansions in Series of Mathieu Functions

… ►Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-infinite strip $S$ that contains the real axis, and $q$ be a normal value (§28.7). … ►

28.11.1 $$f(z)={\alpha }_0{\mathrm{ce}}_0(z,q)+\sum _{n=1}^{\mathrm{\infty }}\left({\alpha }_n{\mathrm{ce}}_n(z,q)+{\beta }_n{\mathrm{se}}_n(z,q)\right),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, ${\alpha }_n$ and ${\beta }_n$

Referenced by:

§28.11

Permalink:

http://dlmf.nist.gov/28.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11 and Ch.28

…

8: 18.18 Sums

… ►

18.18.1 $$a_n=\frac{n!(2n+\alpha +\beta +1)\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{2^{\alpha +\beta +1}\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{\int }_{-1}^{1}f(x)P_n^{(\alpha ,\beta )}\left(x\right){(1-x)}^{\alpha }{(1+x)}^{\beta }dx.$$

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Defines:

$a_n$ (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i), §18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

… ►

18.18.7 $$d_n=\frac{1}{\sqrt{\pi }{2}^{n}n!}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)H_n\left(x\right){\mathrm{e}}^{-x^2}dx.$$

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Defines:

$d_n$ (locally)

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $n$: nonnegative integer, $f(z)$: analytic function and $x$: real variable

Referenced by:

§18.18(i)

Permalink:

http://dlmf.nist.gov/18.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.18(i), §18.18(i), §18.18 and Ch.18

…

9: 35.2 Laplace Transform

… ►

35.2.1 $$g(𝐙)={\int }_{𝛀}\mathrm{etr}\left(-𝐙𝐗\right)f(𝐗)d𝐗,$$

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Defines:

$g$: Laplace transformation of $f$ (locally)

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $𝐙$: complex symmetric $m\times m$ matrix and $f(𝐗)$: complex-valued function of matrix argument

Keywords:

Laplace transform

Referenced by:

§35.2

Permalink:

http://dlmf.nist.gov/35.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

►where the integration variable $𝐗$ ranges over the space $𝛀$. … ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►

35.2.3

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Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

10: 2.4 Contour Integrals

… ►

2.4.10 $$I(z)={\int }_a^b{\mathrm{e}}^{-zp(t)}q(t)dt,$$

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Defines:

$I(z)$: contour integral (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $a$: left endpoint, $b$: right endpoint, $p(t)$: analytic function and $q(t)$: analytic function

Referenced by:

§2.4(iv)

Permalink:

http://dlmf.nist.gov/2.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(iii), §2.4 and Ch.2

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