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1: 28.36 Software

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§28.36(ii) Characteristic Exponents and Eigenvalues

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2: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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3: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number $x$ is represented as …where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ is the most significant bit, $p$ ($\in \mathbb{N}$) is the number of significant bits $b_j$, $b_{p-1}$ is the least significant bit, $E$ is an integer called the exponent, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part. … ►

3.1.2 $${(-1)}^s{2}^E\sum _{j=0}^{p-1}{b}_j{2}^{-j},$$

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Symbols:

$b_p$: bit, $s$: sign, $p$: number of significant bits and $E$: exponent

Permalink:

http://dlmf.nist.gov/3.1.E2

Encodings:

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Annotations for §3.1(i), §3.1 and Ch.3

… ►Let $E_{\min }\le E\le E_{\max }$ with and $E_{\max }>0$. For given values of $E_{\min }$, $E_{\max }$, and $p$, the format width in bits $N$ of a computer word is the total number of bits: the sign (one bit), the significant bits $b_1,b_2,\mathrm{\dots },b_{p-1}$ ($p-1$ bits), and the bits allocated to the exponent (the remaining $N-p$ bits). …

4: 27.3 Multiplicative Properties

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27.3.2 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}f(p_r^{a_r}).$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20, §27.3

Permalink:

http://dlmf.nist.gov/27.3.E2

Encodings:

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Annotations for §27.3 and Ch.27

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27.3.5 $$d\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}(1+a_r),$$

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Symbols:

$d_k\left(n\right)$: divisor function, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer and $a_r$: positive exponent

Permalink:

http://dlmf.nist.gov/27.3.E5

Encodings:

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See also:

Annotations for §27.3 and Ch.27

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

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Annotations for §27.3 and Ch.27

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27.3.10 $$f(n)=\prod _{r=1}^{\nu \left(n\right)}{\left(f(p_r)\right)}^{a_r}.$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers, $a_r$: positive exponent and $f(n)$: multiplicative function

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.3.E10

Encodings:

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See also:

Annotations for §27.3 and Ch.27

5: 36.6 Scaling Relations

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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►For the results in this section and more extensive lists of exponents see Berry (1977) and Varčenko (1976).

6: 31.14 General Fuchsian Equation

… ►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. …

7: 31.3 Basic Solutions

… ► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … ►Similarly, if $\gamma \ne 1,2,3,\mathrm{\dots }$, then the solution of (31.2.1) that corresponds to the exponent $1-\gamma$ at $z=0$ is … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-ϵ$ at $z=a$ are respectively, … ►Solutions of (31.2.1) corresponding to the exponents $\alpha$ and $\beta$ at $z=\mathrm{\infty }$ are respectively, …

8: 28.29 Definitions and Basic Properties

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§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

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28.29.9 $$2\cos \left(\pi \nu \right)=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda ).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\nu$: complex parameter, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Referenced by:

§28.29(ii), §28.29(ii), §28.29(ii)

Permalink:

http://dlmf.nist.gov/28.29.E9

Encodings:

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See also:

Annotations for §28.29(ii), §28.29 and Ch.28

… ►Given $\lambda$ together with the condition (28.29.6), the solutions $\pm \nu$ of (28.29.9) are the characteristic exponents of (28.29.1). …

9: 31.8 Solutions via Quadratures

… ►For half-odd-integer values of the exponent parameters: … ►The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

10: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, …where $p_1,p_2,\mathrm{\dots },p_{\nu \left(n\right)}$ are the distinct prime factors of $n$, each exponent $a_r$ is positive, and $\nu \left(n\right)$ is the number of distinct primes dividing $n$. … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … ►In the following examples, $a_1,\mathrm{\dots },a_{\nu \left(n\right)}$ are the exponents in the factorization of $n$ in (27.2.1). …