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1: 10.28 Wronskians and Cross-Products

§10.28 Wronskians and Cross-Products

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10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

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10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

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Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

2: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

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10.5.1 $$𝒲\left\{J_{\nu }\left(z\right),J_{-\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)J_{-\nu }\left(z\right)+J_{\nu }\left(z\right)J_{-\nu -1}\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.15

Permalink:

http://dlmf.nist.gov/10.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.2 $$𝒲\left\{J_{\nu }\left(z\right),Y_{\nu }\left(z\right)\right\}=J_{\nu +1}\left(z\right)Y_{\nu }\left(z\right)-J_{\nu }\left(z\right)Y_{\nu +1}\left(z\right)=2/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.16

Permalink:

http://dlmf.nist.gov/10.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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3: 10.50 Wronskians and Cross-Products

§10.50 Wronskians and Cross-Products

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$$𝒲\left\{{𝗃}_n\left(z\right),{𝗒}_n\left(z\right)\right\}=z^{-2},$$

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$$𝒲\left\{{𝗁}_n^{(1)}\left(z\right),{𝗁}_n^{(2)}\left(z\right)\right\}=-2\mathrm{i}{z}^{-2}.$$

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$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗂}_n^{(2)}\left(z\right)\right\}={(-1)}^{n+1}{z}^{-2},$$

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$$𝒲\left\{{𝗂}_n^{(1)}\left(z\right),{𝗄}_n\left(z\right)\right\}=𝒲\left\{{𝗂}_n^{(2)}\left(z\right),{𝗄}_n\left(z\right)\right\}=-\frac{1}{2}\pi z^{-2}.$$

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4: 14.2 Differential Equations

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§14.2(iv) Wronskians and Cross-Products

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14.2.3 $$𝒲\left\{{𝖯}_{\nu }^{-\mu }\left(x\right),{𝖯}_{\nu }^{-\mu }\left(-x\right)\right\}=\frac{2}{\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +\mu +1\right)\left(1-x^2\right)},$$

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Symbols:

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

Referenced by:

§14.20(i)

Permalink:

http://dlmf.nist.gov/14.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.6 $$𝒲\left\{{𝖯}_{\nu }^{-\mu }\left(x\right),{𝖰}_{\nu }^{\mu }\left(x\right)\right\}=\frac{\cos \left(\mu \pi \right)}{1-x^2},$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.7 $$𝒲\left\{P_{\nu }^{-\mu }\left(x\right),P_{\nu }^{\mu }\left(x\right)\right\}=𝒲\left\{{𝖯}_{\nu }^{-\mu }\left(x\right),{𝖯}_{\nu }^{\mu }\left(x\right)\right\}=\frac{2\sin \left(\mu \pi \right)}{\pi \left(1-x^2\right)},$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

(14.2.7), §14.21(iii), Erratum (V1.0.9) for Equation (14.2.7)

Permalink:

http://dlmf.nist.gov/14.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.9):

(14.2.7) has been expanded to provide the Wronskian for Ferrers functions as well as for associated Legendre functions.

Suggested 2014-05-22 by Howard Cohl

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.10 $$𝒲\left\{P_{\nu }^{\mu }\left(x\right),Q_{\nu }^{\mu }\left(x\right)\right\}=-{\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(x^2-1\right)},$$

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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, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.1.8 (modified)

Permalink:

http://dlmf.nist.gov/14.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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5: 9.11 Products

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§9.11(ii) Wronskian

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9.11.2 $$𝒲\left\{{\mathrm{Ai}}^2\left(z\right),\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right),{\mathrm{Bi}}^2\left(z\right)\right\}=2{\pi }^{-3}.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Proof sketch:

Use (9.2.1) and (9.2.7).

Permalink:

http://dlmf.nist.gov/9.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(ii), §9.11 and Ch.9

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9.11.6 $$\int w_1{w}_2^{\prime }dz=\frac{1}{2}\left(w_1{w}_2+z𝒲\left\{w_1,w_2\right\}\right),$$

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Symbols:

$𝒲$: Wronskian, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: function

Source:

Albright (1977, (A.17))

Permalink:

http://dlmf.nist.gov/9.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

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9.11.9 $$\int z{w}_1{w}_2^{\prime }dz=\frac{1}{2}{w}_1^{\prime }{w}_2^{\prime }+\frac{1}{4}{z}^2𝒲\left\{w_1,w_2\right\},$$

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Symbols:

$𝒲$: Wronskian, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: function

Source:

Albright (1977, (A.20))

Permalink:

http://dlmf.nist.gov/9.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

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9.11.11 $$\int \frac{1}{w_1^2}{f}^{\prime }\left(\frac{w_2}{w_1}\right)dz=\frac{1}{𝒲\left\{w_1,w_2\right\}}f\left(\frac{w_2}{w_1}\right).$$

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Symbols:

$𝒲$: Wronskian, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $w$: function and $f$: function

Source:

Albright and Gavathas (1986, p. 2664)

Permalink:

http://dlmf.nist.gov/9.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(iv), §9.11 and Ch.9

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6: 1.13 Differential Equations

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Wronskian

►The Wronskian of $w_1(z)$ and $w_2(z)$ is defined by …(More generally $𝒲\left\{w_1(z),\mathrm{\dots },w_n(z)\right\}=det\left[w_k^{(j-1)}(z)\right]$, where $1\le j,k\le n$.) …If $f(z)=0$, then the Wronskian is constant. ►The following three statements are equivalent: $w_1(z)$ and $w_2(z)$ comprise a fundamental pair in $D$; $𝒲\left\{w_1(z),w_2(z)\right\}$ does not vanish in $D$; $w_1(z)$ and $w_2(z)$ are linearly independent, that is, the only constants $A$ and $B$ such that …

7: 9.2 Differential Equation

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§9.2(iv) Wronskians

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9.2.7 $$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right)\right\}=\frac{1}{\pi },$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Source:

Olver (1997b, (1.19), p. 393)

A&S Ref:

10.4.10

Referenced by:

(9.10.2), (9.10.3), (9.11.2), (9.8.13), (9.8.17), (9.9.3), (9.9.4)

Permalink:

http://dlmf.nist.gov/9.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

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9.2.8 $$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)\right\}=\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}/6}}{2\pi },$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Derivable from Olver (1997b, p. 416).

A&S Ref:

10.4.11 10.4.12

Permalink:

http://dlmf.nist.gov/9.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

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9.2.9 $$𝒲\left\{\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)\right\}=\frac{1}{2\pi \mathrm{i}}.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Proof sketch:

Derivable from Olver (1997b, p. 416).

A&S Ref:

10.4.13

Permalink:

http://dlmf.nist.gov/9.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(iv), §9.2 and Ch.9

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8: 30.5 Functions of the Second Kind

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30.5.4 $$𝒲\left\{{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),{\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\right\}=\frac{(n+m)!}{(1-x^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)(\ne 0),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.5 and Ch.30

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9: 12.2 Differential Equations

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§12.2(iii) Wronskians

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12.2.10 $$𝒲\left\{U(a,z),V(a,z)\right\}=\sqrt{2/\pi },$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.4.1

Permalink:

http://dlmf.nist.gov/12.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iii), §12.2 and Ch.12

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12.2.11 $$𝒲\left\{U(a,z),U(a,-z)\right\}=\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iii), §12.2 and Ch.12

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12.2.12 $$𝒲\left\{U(a,z),U(-a,\pm \mathrm{i}z)\right\}=\mp \mathrm{i}{\mathrm{e}}^{\pm \mathrm{i}\pi (\frac{1}{2}a+\frac{1}{4})}.$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.4.1

Permalink:

http://dlmf.nist.gov/12.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §12.2(iii), §12.2 and Ch.12

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10: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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Wronskians

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28.5.8 $$𝒲\left\{{\mathrm{ce}}_n,{\mathrm{fe}}_n\right\}={\mathrm{ce}}_n(0,q){\mathrm{fe}}_n^{\prime }(0,q),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $𝒲$: Wronskian, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.9 $$𝒲\left\{{\mathrm{se}}_n,{\mathrm{ge}}_n\right\}=-{\mathrm{se}}_n^{\prime }(0,q){\mathrm{ge}}_n(0,q).$$

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Symbols:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, ${\mathrm{se}}_n(z,q)$: Mathieu function, $𝒲$: Wronskian, $q=h^2$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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