Wronskian

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21: 32.10 Special Function Solutions

… ►where ${\tau }_n(z)$ is the $n\times n$ Wronskian determinant ►

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

ⓘ

Symbols:

$𝒲$: Wronskian, $n$: integer, $z$: real, $\varphi (z)$: function and ${\tau }_n(z)$: determinant

Referenced by:

§32.10(ii), Erratum (V1.2.0) for Equations (32.8.10), (32.10.9)

Permalink:

http://dlmf.nist.gov/32.10.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.2.0):

The right-hand side of this equation, which was originally written as a matrix determinant, was rewritten using the Wronskian determinant notation.

See also:

Annotations for §32.10(ii), §32.10 and Ch.32

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22: 10.45 Functions of Imaginary Order

… ►

10.45.4 $$𝒲\left\{{\tilde{K}}_{\nu }\left(x\right),{\tilde{I}}_{\nu }\left(x\right)\right\}=1/x.$$

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Symbols:

$𝒲$: Wronskian, ${\tilde{I}}_{\nu }\left(x\right)$: modified Bessel function of the first kind of imaginary order, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

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23: 30.11 Radial Spheroidal Wave Functions

… ►

§30.11(iv) Wronskian

►

30.11.7 $$𝒲\left\{S_n^{m(1)}(z,\gamma ),S_n^{m(2)}(z,\gamma )\right\}=\frac{1}{\gamma (z^2-1)}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iv), §30.11 and Ch.30

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24: 15.10 Hypergeometric Differential Equation

… ►

Singularity $z=0$

… ►

15.10.3 $$𝒲\left\{f_1(z),f_2(z)\right\}=(1-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

►

Singularity $z=1$

… ►

15.10.5 $$𝒲\left\{f_1(z),f_2(z)\right\}=(a+b-c)z^{-c}{(1-z)}^{c-a-b-1}.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $f_1(z)$, $f_2(z)$: fundamental solutions

Permalink:

http://dlmf.nist.gov/15.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

►

Singularity $z=\mathrm{\infty }$

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25: 33.23 Methods of Computation

… ►Combined with the Wronskians (33.2.12), the values of $F_{\mathrm{\ell }}$, $G_{\mathrm{\ell }}$, and their derivatives can be extracted. …

26: 10.24 Functions of Imaginary Order

… ►

10.24.5 $$𝒲\left\{{\tilde{J}}_{\nu }\left(x\right),{\tilde{Y}}_{\nu }\left(x\right)\right\}=2/(\pi x).$$

ⓘ

Symbols:

${\tilde{J}}_{\nu }\left(x\right)$: Bessel function of imaginary order, ${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.24

Permalink:

http://dlmf.nist.gov/10.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

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27: 14.20 Conical (or Mehler) Functions

… ►

§14.20(i) Definitions and Wronskians

… ►

14.20.4 $$𝒲\left\{{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right),{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right)\right\}=\frac{2}{{|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)|}^2(1-x^2)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $𝒲$: Wronskian, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

►

14.20.5 $$𝒲\left\{{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right),{\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)\right\}=\frac{\pi ({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{{|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)|}^2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))(1-x^2)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

… ►

14.20.6 $$P_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\mathrm{i}{\mathrm{e}}^{-\mu \pi \mathrm{i}}}{\sinh \left(\tau \pi \right){\left|\mathrm{\Gamma }\left(\mu +\frac{1}{2}+\mathrm{i}\tau \right)\right|}^2}\left(Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)-Q_{-\frac{1}{2}-\mathrm{i}\tau }^{\mu }\left(x\right)\right),$$ $\tau \ne 0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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28: 18.36 Miscellaneous Polynomials

… ►

18.36.9 $${\widehat{H}}_{n+3}\left(x\right)=\frac{(4x^2+2)H_{n+1}\left(x\right)+8x{H}_n\left(x\right)}{{\pi }^{1/4}\sqrt{2^{n+1}(n+3)n!}}=\frac{𝒲\left\{H_1\left(x\right),H_2\left(x\right),H_{n+3}\left(x\right)\right\}}{{\pi }^{1/4}\sqrt{2^{n+7}(n+1)(n+2)(n+3)!}},$$ $n=0,1,\mathrm{\dots }$,

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, ${\widehat{H}}_n\left(x\right)$: exceptional Hermite polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.39(i)

Permalink:

http://dlmf.nist.gov/18.36.E9

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.36(vi), §18.36(vi), §18.36 and Ch.18

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29: 12.14 The Function $W(a,x)$

… ►

§12.14(ii) Values at $z=0$ and Wronskian

… ►

12.14.3 $$𝒲\left\{W(a,x),W(a,-x)\right\}=1.$$

ⓘ

Symbols:

$𝒲$: Wronskian, $W(a,x)$: parabolic cylinder function, $x$: real variable and $a$: real or complex parameter

A&S Ref:

19.18.1

Permalink:

http://dlmf.nist.gov/12.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ii), §12.14 and Ch.12

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30: 31.8 Solutions via Quadratures

… ►Lastly, ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$, are the zeros of the Wronskian of $w_{+}(𝐦;\lambda ;z)$ and $w_{-}(𝐦;\lambda ;z)$. …