fundamental%20theorem%20of%20arithmetic

SearchAdvancedHelp

(0.002 seconds)

11—20 of 238 matching pages

11: 28.29 Definitions and Basic Properties

… ►

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

… ►

28.29.4 $$w_{\text{I}}(z+\pi ,\lambda )=w_{\text{I}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{I}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E4

Encodings:

pMML, png, TeX

See also:

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

►

28.29.5 $$w_{\text{II}}(z+\pi ,\lambda )=w_{\text{II}}(\pi ,\lambda )w_{\text{I}}(z,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )w_{\text{II}}(z,\lambda ).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E5

Encodings:

pMML, png, TeX

See also:

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

… ►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. … ►

28.29.15 $$△(\lambda )=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.29.E15

Encodings:

pMML, png, TeX

See also:

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

…

12: 8 Incomplete Gamma and Related
Functions

…

13: 28 Mathieu Functions and Hill’s Equation

…

14: 28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions $w_{\text{I}}(z;a,q),w_{\text{II}}(z;a,q)$ called basic solutions with … ►

28.2.6 $$𝒲\left\{w_{\text{I}},w_{\text{II}}\right\}=1,$$

ⓘ

Symbols:

$𝒲$: Wronskian, $w_{\text{I}}(z;a,q)$: fundamental solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.29(ii)

Permalink:

http://dlmf.nist.gov/28.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

… ►

§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

… ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

15: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 23 Weierstrass Elliptic and Modular
Functions

…

17: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …

18: 36 Integrals with Coalescing Saddles

…

19: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

20: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …