Wronskian

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31: 9.13 Generalized Airy Functions

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9.13.8 $$𝒲\left\{A_n\left(z\right),B_n\left(z\right)\right\}=\frac{2}{\pi }{p}^{1/2}\sin \left(p\pi \right).$$

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Symbols:

$A_n\left(z\right)$: generalized Airy function, $B_n\left(z\right)$: generalized Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable, $n$: parameter and $p$: variable

Source:

Swanson and Headley (1967, p. 1403)

Permalink:

http://dlmf.nist.gov/9.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►These properties include Wronskians, asymptotic expansions, and information on zeros. …

32: 28.2 Definitions and Basic Properties

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

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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

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33: 10.15 Derivatives with Respect to Order

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34: Errata

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32.8.10 $${\tau }_n(z)=𝒲\left\{p_1(z),p_3(z),\mathrm{\dots },p_{2n-1}(z)\right\}$$

32.10.9 $${\tau }_n(z)=𝒲\left\{\varphi (z),{\varphi }^{\prime }(z),\mathrm{\dots },{\varphi }^{(n-1)}(z)\right\}$$

The right-hand side of these equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation. Also, in each preceding sentence, the word ‘determinant’ was replaced with ‘Wronskian determinant’.

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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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The Wronskian was generalized to include both associated Legendre and Ferrers functions.

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