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DLMF: §10.28 Wronskians and Cross-Products ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Modified Bessel Functions 10.27 Connection Formulas 10.29 Recurrence Relations and Derivatives

§10.28 Wronskians and Cross-Products

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Keywords:: Wronskians, cross-products, modified Bessel functions
Notes:: For the Wronskians use (1.13.5) and the limiting forms in §10.30. For the cross-products apply (10.29.2).
Permalink:: http://dlmf.nist.gov/10.28
See also:: Annotations for Ch.10

10.28.1		$\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.14 Permalink: http://dlmf.nist.gov/10.28.E1 Encodings: TeX, pMML, png See also: Annotations for §10.28 and Ch.10

10.28.2		$\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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: TeX, pMML, png See also: Annotations for §10.28 and Ch.10

10.27 Connection Formulas 10.29 Recurrence Relations and Derivatives

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