analytical properties

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11: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. … ►

§4.2(iii) The Exponential Function

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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

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12: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

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§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

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13: 14.21 Definitions and Basic Properties

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14: 25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

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15: 23.15 Definitions

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16: 25.12 Polylogarithms

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§25.12(i) Dilogarithms

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§25.12(ii) Polylogarithms

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17: 25.2 Definition and Expansions

§25.2 Definition and Expansions

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18: 28.12 Definitions and Basic Properties

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§28.12(iii) Functions ${\mathrm{ce}}_{\nu }(z,q)$, ${\mathrm{se}}_{\nu }(z,q)$, when $\nu \notin \mathbb{Z}$

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19: 21.2 Definitions

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21.2.1 $$\theta \left(\mathbf{z}\right|\mathbf{\Omega })=\sum _{\mathbf{n}\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}\mathbf{n}\cdot \mathbf{\Omega }\cdot \mathbf{n}+\mathbf{n}\cdot \mathbf{z}\right)}.$$

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Defines:

$\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $\mathbf{\Omega }$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(i), §21.2 and Ch.21

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20: 16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. …