Heun functions

SearchAdvancedHelp

(0.001 seconds)

11—20 of 38 matching pages

11: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. …

12: Vadim B. Kuznetsov

… ►

31 Heun Functions

13: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

14: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … ►For Heun functions (§31.4) they are convergent inside the ellipse $ℰ$. Every Heun function can be represented by a series of Type II. … ►

15: Brian D. Sleeman

… ►

31 Heun Functions

…

16: 31.8 Solutions via Quadratures

… ►

$ϵ=-m_3+\frac{1}{2}$ , $m_0,m_1,m_2,m_3=0,1,2,\mathrm{\dots }$,

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

17: 31.16 Mathematical Applications

§31.16 Mathematical Applications

►

§31.16(i) Uniformization Problem for Heun’s Equation

… ►

31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

ⓘ

Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

18: 31.3 Basic Solutions

… ►

31.3.1 $$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)=\sum _{j=0}^{\mathrm{\infty }}{c}_j{z}^j,$$ ,

ⓘ

Defines:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $c_j$: coefficients

Referenced by:

§31.3(iii), §31.7(ii)

Permalink:

http://dlmf.nist.gov/31.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(i), §31.3 and Ch.31

… ►

31.3.6 $$H\mathrm{\ell }(1-a,\alpha \beta -q;\alpha ,\beta ,\delta ,\gamma ;1-z),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

… ►

31.3.8 $$H\mathrm{\ell }(\frac{a}{a-1},\frac{\alpha \beta a-q}{a-1};\alpha ,\beta ,ϵ,\delta ;\frac{a-z}{a-1}),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\delta$: 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

Permalink:

http://dlmf.nist.gov/31.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

… ►

31.3.12 $$H\mathrm{\ell }(1/a,q/a;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma -\delta ;z/a),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(iii), §31.3 and Ch.31

… ►

31.3.13 $${(1-z)}^{-\alpha }H\mathrm{\ell }(\frac{a}{a-1},-\frac{q-a\alpha \gamma }{a-1};\alpha ,\alpha +1-\delta ,\gamma ,\alpha +1-\beta ;\frac{z}{z-1}),$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §31.3(iii), §31.3 and Ch.31

…

19: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

ⓘ

Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

…

20: 31.14 General Fuchsian Equation

… ►The algorithm returns a list of solutions if they exist. …