equation of Ince

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1: 28.31 Equations of Whittaker–Hill and Ince

§28.31 Equations of Whittaker–Hill and Ince

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§28.31(ii) Equation of Ince; Ince Polynomials

… ►The result is the Equation of Ince: … ►

28.31.5 $$w_{o,s}(z)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{B}_{2\mathrm{\ell }+s}\sin (2\mathrm{\ell }+s)z,$$ $s=1,2$,

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Symbols:

$\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $B$: constant

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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28.31.16 $${\mathrm{ℎ𝑐}}_p^m(z,\xi )={\mathrm{e}}^{-\frac{1}{4}\xi \cos \left(2z\right)}{C}_p^m(z,\xi ),$$

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Defines:

${\mathrm{ℎ𝑐}}_p^m(z,\xi )$: paraboloidal wave function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $m$: integer, $z$: complex variable and $C_p^m(z,\xi )$: Ince polynomials

Permalink:

http://dlmf.nist.gov/28.31.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(iii), §28.31 and Ch.28

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2: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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§28.5(i) Definitions

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3: Bibliography I

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Ordinary Differential Equations. Longmans, Green and Co., London.

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Notes:

Reprinted by Dover Publications, New York, 1944, 1956.

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.13(i), §1.13(i), §1.13(i), §31.12, §31.14(i), §32.2(i), §32.2(vi), Erratum (V1.0.25) for Section 1.13.

Encodings:

BibTeX

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4: 28.2 Definitions and Basic Properties

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5: 31.14 General Fuchsian Equation

§31.14 General Fuchsian Equation

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§31.14(i) Definitions

… ►Heun’s equation (31.2.1) corresponds to $N=3$. ►

Normal Form

… ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …

6: 1.13 Differential Equations

… ►(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) …

7: 32.2 Differential Equations

§32.2 Differential Equations

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§32.2(i) Introduction

►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►

§32.2(ii) Renormalizations

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8: 31.12 Confluent Forms of Heun’s Equation

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Confluent Heun Equation

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Doubly-Confluent Heun Equation

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Biconfluent Heun Equation

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Triconfluent Heun Equation

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9: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … … ►satisfies the continued-fraction equation … ►The quantity $H=2a_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with … ►The quantity $H=2b_{\nu }^{2m+1}\left(k^2\right)-\nu (\nu +1)k^2$ satisfies equation (29.3.10) with …

10: Errata

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In Equation (1.13.4), the determinant form of the two-argument Wronskian

1.13.4

was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the $n$-argument Wronskian is given by $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$. Immediately below Equation (1.13.4), a sentence was added giving the definition of the $n$-argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for $n$th-order differential equations. A reference to Ince (1926, §5.2) was added.

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