Laguere EOP’s

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1: 7.20 Mathematical Applications

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§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. … ►

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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

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2: 31.2 Differential Equations

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§31.2(i) Heun’s Equation

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

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See also:

Annotations for §31.2(i), §31.2 and Ch.31

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Jacobi’s Elliptic Form

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Weierstrass’s Form

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§31.2(v) Heun’s Equation Automorphisms

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3: 29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

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

4: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

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28.2.1 $$w^{\prime \prime }+(a-2q\cos \left(2z\right))w=0.$$

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Symbols:

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

A&S Ref:

20.3.1 20.1.1 (in slightly different form)

Referenced by:

§1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)

Permalink:

http://dlmf.nist.gov/28.2.E1

Encodings:

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See also:

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

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

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§28.2(iv) Floquet Solutions

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5: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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

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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:

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See also:

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

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

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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:

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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. … ►

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

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6: 28.20 Definitions and Basic Properties

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

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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:

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See also:

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

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

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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

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§22.16(ii) Jacobi’s Epsilon Function

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Integral Representations

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§22.16(iii) Jacobi’s Zeta Function

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Definition

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Properties

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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: 18.36 Miscellaneous Polynomials

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§18.36(vi) Exceptional Orthogonal Polynomials

… ►EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. … ►

Type I $X_1$-Laguerre EOP’s

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Type III $X_2$-Hermite EOP’s

►Hermite EOP’s are defined in terms of classical Hermite OP’s. …

10: 18.38 Mathematical Applications

… ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …Hermite EOP’s appear in solutions of a rationally modified Schrödinger equation in §18.39. Much of the exploration of the EOP’s is based on the operator algebra as developed in SUSY, above. ►

EOP’s, Painlevé Transcendents, and Quantum Mechanics

►EOP’s are the subject of recent work on rational solutions to the fourth Painlevé equation, see Clarkson (2003a) and Marquette and Quesne (2016),where use of Hermite EOP’s makes a connection to quantum mechanics. …