branch conventions

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1: 10.2 Definitions

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Branch Conventions

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2: 10.25 Definitions

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Branch Conventions

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3: 4.12 Generalized Logarithms and Exponentials

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4.12.9 $$\psi (x)=\mathrm{\ell }+\underset{\mathrm{\ell }\text{ times}}{\underbrace{\ln \mathrm{\cdots }\ln }}x,$$ $x>1$,

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Symbols:

$\ln z$: principal branch of logarithm function, $x$: real variable, $\psi (x)$: generalized logarithm and $\mathrm{\ell }$: positive integer

Permalink:

http://dlmf.nist.gov/4.12.E9

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

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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

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4: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. ►The principal value, or principal branch, is defined by … ►Most texts extend the definition of the principal value to include the branch cut … ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. … ►With this convention, …

5: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►For example, for the hypergeometric function we often use the notation $𝐅(a,b;c;z)$ (§15.2(i)) in place of the more conventional ${}_2{}^{}F_1^{}(a,b;c;z)$ or $F(a,b;c;z)$. … ►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … ►Another numerical convention is that decimals followed by dots are unrounded; without the dots they are rounded. …