exponent%20pairs

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1: 36.5 Stokes Sets

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36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). …

2: 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 $n$. …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►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). …

3: 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). … ►If $\nu$ $(\ne 0,1)$ is a solution of (28.29.9), then $F_{\nu }(z)$, $F_{-\nu }(z)$ comprise a fundamental pair of solutions of Hill’s equation. …

4: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. … ►Their product $m$ has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

5: 15.11 Riemann’s Differential Equation

… ►Here $\{a_1,a_2\}$, $\{b_1,b_2\}$, $\{c_1,c_2\}$ are the exponent pairs at the points $\alpha$, $\beta$, $\gamma$, respectively. Cases in which there are fewer than three singularities are included automatically by allowing the choice $\{0,1\}$ for exponent pairs. …

6: 28.36 Software

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§28.36(ii) Characteristic Exponents and Eigenvalues

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7: 14.2 Differential Equations

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§14.2(iii) Numerically Satisfactory Solutions

►Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\mathrm{\infty }$, with exponent pairs $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, and $\{\nu +1,-\nu \}$, respectively; compare §2.7(i). … ►Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval . When $\mu -\nu =0,-1,-2,\mathrm{\dots }$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${𝖯}_{\nu }^{-\mu }\left(x\right)$ and ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair. … ►Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval . …

8: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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9: 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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See also:

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). …

10: 15.10 Hypergeometric Differential Equation

… ►It has regular singularities at $z=0,1,\mathrm{\infty }$, with corresponding exponent pairs $\{0,1-c\}$, $\{0,c-a-b\}$, $\{a,b\}$, respectively. When none of the exponent pairs differ by an integer, that is, when none of $c$, $c-a-b$, $a-b$ is an integer, we have the following pairs $f_1(z)$, $f_2(z)$ of fundamental solutions. … ►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …