analytic properties

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1: 5.2 Definitions

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

Encodings:

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See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

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See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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2: 28.7 Analytic Continuation of Eigenvalues

§28.7 Analytic Continuation of Eigenvalues

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28.7.4 $$\sum _{n=0}^{\mathrm{\infty }}\left(b_{2n+2}\left(q\right)-{(2n+2)}^2\right)=0.$$

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Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.7.E4

Encodings:

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See also:

Annotations for §28.7 and Ch.28

3: 16.2 Definition and Analytic Properties

§16.2 Definition and Analytic Properties

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§16.2(ii) Case $p\le q$

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§16.2(iii) Case $p=q+1$

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§16.2(iv) Case $p>q+1$

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§16.2(v) Behavior with Respect to Parameters

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4: William P. Reinhardt

… ►Reinhardt is a theoretical chemist and atomic physicist, who has always been interested in orthogonal polynomials and in the analyticity properties of the functions of mathematical physics. …

5: 4.14 Definitions and Periodicity

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4.14.7 $$\cot z=\frac{\cos z}{\sin z}=\frac{1}{\tan z}.$$

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Defines:

$\cot z$: cotangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.6

Referenced by:

§4.14, §4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.14.E7

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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6: 15.2 Definitions and Analytical Properties

§15.2 Definitions and Analytical Properties

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§15.2(ii) Analytic Properties

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …

7: 4.28 Definitions and Periodicity

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Periodicity and Zeros

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8: 35.2 Laplace Transform

… ►where the integration variable $𝐗$ ranges over the space $𝛀$. …

9: 22.17 Moduli Outside the Interval [0,1]

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§22.17(ii) Complex Moduli

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10: 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:

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See also:

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

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