Olver%E2%80%99s

SearchAdvancedHelp

(0.003 seconds)

1—10 of 655 matching pages

1: 7.20 Mathematical Applications

… ►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). … ►

§7.20(ii) Cornu’s Spiral

►Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\ge 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. … ►

►►

Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by $x=C\left(t\right)$, $y=S\left(t\right)$, $t\in [0,\mathrm{\infty })$. Magnify

…

2: 31.2 Differential Equations

… ►

§31.2(i) Heun’s Equation

… ►

Jacobi’s Elliptic Form

… ►

Weierstrass’s Form

… ►

§31.2(v) Heun’s Equation Automorphisms

… ►

Composite Transformations

…

3: 29.2 Differential Equations

… ►

§29.2(i) Lamé’s Equation

… ►

§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). … ►

4: 7.2 Definitions

… ►

§7.2(ii) Dawson’s Integral

►

7.2.5 $$F\left(z\right)={\mathrm{e}}^{-z^2}{\int }_0^z{\mathrm{e}}^{t^2}dt.$$

ⓘ

Defines:

$F\left(z\right)$: Dawson’s integral

Symbols:

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

A&S Ref:

7.1.1

Permalink:

http://dlmf.nist.gov/7.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(ii), §7.2 and Ch.7

… ►

7.2.8 $$S\left(z\right)={\int }_0^z\sin \left(\frac{1}{2}\pi t^2\right)dt,$$

ⓘ

Defines:

$S\left(z\right)$: Fresnel integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.2

Referenced by:

§7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

► $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. … ►

$\underset{x\to \mathrm{\infty }}{lim}S\left(x\right)=\frac{1}{2}.$

…

5: 28.2 Definitions and Basic Properties

… ►

§28.2(i) Mathieu’s Equation

… ►

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

… ►This is the characteristic equation of Mathieu’s equation (28.2.1). … ►

§28.2(iv) Floquet Solutions

… ► …

6: 28.20 Definitions and Basic Properties

… ►

§28.20(i) Modified Mathieu’s Equation

►When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: ►

28.20.1 $$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

A&S Ref:

20.1.2 (in slightly different form) 20.8.6

Referenced by:

§28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)

Permalink:

http://dlmf.nist.gov/28.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►

28.20.2 $$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$ $\zeta =\cosh z$.

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution

Referenced by:

§28.20(iii)

Permalink:

http://dlmf.nist.gov/28.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(i), §28.20 and Ch.28

… ►For $s\in \mathbb{Z}$, …

7: 22.16 Related Functions

… ►

§22.16(ii) Jacobi’s Epsilon Function

►

Integral Representations

… ►

§22.16(iii) Jacobi’s Zeta Function

►

Definition

… ►

Properties

…

8: 19.2 Definitions

… ►Because $s^2$ is a polynomial, we have … ►

§19.2(ii) Legendre’s Integrals

… ►Legendre’s complementary complete elliptic integrals are defined via … ►

§19.2(iii) Bulirsch’s Integrals

►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …

9: 2 Asymptotic Approximations

… …

10: Frank W. J. Olver

Profile
Frank W. J. Olver

… ►Frank W. J. Olver (b. … ►Olver was elected a Fellow of the U. … ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. … ►Fitting capstones to Olver’s long career, these works are on track to extend the legacy of the classic Abramowitz and Stegun handbook well into the 21st century. …