locally analytic

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11: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ► … ►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: … ►

§3.8(v) Zeros of Analytic Functions

… ►The rule converges locally and is cubically convergent. …

12: 1.10 Functions of a Complex Variable

… ► ►

§1.10(ii) Analytic Continuation

… ► … ►

Schwarz Reflection Principle

… ►

Analytic Functions

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13: 21.7 Riemann Surfaces

… ►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemann surface. All compact Riemann surfaces can be obtained this way. ►Since a Riemann surface $\mathrm{\Gamma }$ is a two-dimensional manifold that is orientable (owing to its analytic structure), its only topological invariant is its genus $g$ (the number of handles in the surface). … ►If a local coordinate $z$ is chosen on the Riemann surface, then the local coordinate representation of these holomorphic differentials is given by ►

21.7.4 $${\omega }_j=f_j(z)dz,$$ $j=1,2,\mathrm{\dots },g$,

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Symbols:

$dx$: differential of $x$ , $g$: positive integer, $\omega$: differential and $f_j(z)$: analytic functions

Permalink:

http://dlmf.nist.gov/21.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(i), §21.7 and Ch.21

►where $f_j(z)$, $j=1,2,\mathrm{\dots },g$ are analytic functions. …

14: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. …

15: 2.6 Distributional Methods

… ►Let $f(t)$ be locally integrable on $[0,\mathrm{\infty })$. The Stieltjes transform of $f(t)$ is defined by … ► $ℳf\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). … ►In terms of the convolution product … ►where $ℳf\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …

16: 4.12 Generalized Logarithms and Exponentials

… ►

4.12.10

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Defines:

$\mathrm{\ell }$: positive integer (locally)

Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.12.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.24):

The $\mathrm{\ell }$-times repeated application of $\ln$, previously given as ${\ln }^{(\mathrm{\ell })}$, has been rewritten using a more conventional notation.

See also:

Annotations for §4.12 and Ch.4

… ►For analytic generalized logarithms, see Kneser (1950).

17: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►Let $q$ be a normal value (§28.12(i)) with respect to $\nu$, and $f(z)$ be a function that is analytic on a doubly-infinite open strip $S$ that contains the real axis. … ►

28.19.2 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{\mathrm{me}}_{\nu +2n}(z,q),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_n$: coefficients

Referenced by:

§28.19

Permalink:

http://dlmf.nist.gov/28.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

… ►

28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

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Defines:

$f_n$: coefficients (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

…

18: 8.21 Generalized Sine and Cosine Integrals

… ►

8.21.2 $$\mathrm{Ci}(a,z)\pm \mathrm{i}\mathrm{Si}(a,z)={\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}a}\gamma (a,z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}).$$

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Defines:

$\mathrm{Ci}(a,z)$: generalized cosine integral and $\mathrm{Si}(a,z)$: generalized sine integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.21(ii)

Permalink:

http://dlmf.nist.gov/8.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(i), §8.21 and Ch.8

… ►Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). … ►

8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

►

8.21.19 $$g(a,z)=\mathrm{si}(a,z)\sin z+\mathrm{ci}(a,z)\cos z.$$

ⓘ

Defines:

$g(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.7 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(vii), §8.21 and Ch.8

…

19: 9.2 Differential Equation

… ►

9.2.1 $$\frac{d^{2}w}{{dz}^2}=zw.$$

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Defines:

$w$: ODE solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

Source:

Olver (1997b, (1.01), p. 392)

A&S Ref:

10.4.1 (in slightly different form)

Referenced by:

§36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

…

20: 4.2 Definitions

… ►

4.2.27 $$z^a=\underset{n\text{ times}}{\underbrace{z\cdot z\mathrm{\cdots }z}}=1/z^{-a}.$$

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Defines:

$a=n$: integer (locally)

Symbols:

$n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

…