10 Bessel Functions Bessel and Hankel Functions 10.3 Graphics 10.5 Wronskians and Cross-Products

§10.4 Connection Formulas

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Keywords:: Bessel functions, Hankel functions, connection formulas
Notes:: See Olver (1997b, pp. 56, 238–239, 242–243) and Watson (1944, pp. 74–75).
Referenced by:: §10.74(i)
Permalink:: http://dlmf.nist.gov/10.4
See also:: Annotations for Ch.10

Other solutions of (10.2.1) include $J_{-\nu}\left(z\right)$ , $Y_{-\nu}\left(z\right)$ , ${H^{(1)}_{-\nu}}\left(z\right)$ , and ${H^{(2)}_{-\nu}}\left(z\right)$ .

10.4.1	$\displaystyle J_{-n}\left(z\right)$	$\displaystyle=(-1)^{n}J_{n}\left(z\right),$
10.4.1	$\displaystyle Y_{-n}\left(z\right)$	$\displaystyle=(-1)^{n}Y_{n}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $n$ : integer and $z$ : complex variable A&S Ref: 9.1.5 Referenced by: §10.7(i), §10.8, §10.8 Permalink: http://dlmf.nist.gov/10.4.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.4 and Ch.10

10.4.2	$\displaystyle{H^{(1)}_{-n}}\left(z\right)$	$\displaystyle=(-1)^{n}{H^{(1)}_{n}}\left(z\right),$
10.4.2	$\displaystyle{H^{(2)}_{-n}}\left(z\right)$	$\displaystyle=(-1)^{n}{H^{(2)}_{n}}\left(z\right).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.4.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.4 and Ch.10

10.4.3	$\displaystyle{H^{(1)}_{\nu}}\left(z\right)$	$\displaystyle=J_{\nu}\left(z\right)+iY_{\nu}\left(z\right),$
10.4.3	$\displaystyle{H^{(2)}_{\nu}}\left(z\right)$	$\displaystyle=J_{\nu}\left(z\right)-iY_{\nu}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.3, 9.1.4 Referenced by: §10.11, §10.19(i), §10.43(ii), §10.44(iii), §10.47(iv), §10.7(i), §10.7(i), §10.8, (9.8.10), (9.8.11), (9.8.12), (9.8.9) Permalink: http://dlmf.nist.gov/10.4.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.4 and Ch.10

10.4.4	$\displaystyle J_{\nu}\left(z\right)$	$\displaystyle=\frac{1}{2}\left({H^{(1)}_{\nu}}\left(z\right)+{H^{(2)}_{\nu}}% \left(z\right)\right),$
10.4.4	$\displaystyle Y_{\nu}\left(z\right)$	$\displaystyle=\frac{1}{2i}\left({H^{(1)}_{\nu}}\left(z\right)-{H^{(2)}_{\nu}}% \left(z\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.17(i), §10.17(iv), §10.27, §10.41(v) Permalink: http://dlmf.nist.gov/10.4.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.4 and Ch.10

10.4.5		$J_{\nu}\left(z\right)=\csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-Y_{% \nu}\left(z\right)\cos\left(\nu\pi\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cos\NVar{z}$ : cosine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.4 Permalink: http://dlmf.nist.gov/10.4.E5 Encodings: TeX, pMML, png See also: Annotations for §10.4 and Ch.10

10.4.6	$\displaystyle{H^{(1)}_{-\nu}}\left(z\right)$	$\displaystyle=e^{\nu\pi i}{H^{(1)}_{\nu}}\left(z\right),$
10.4.6	$\displaystyle{H^{(2)}_{-\nu}}\left(z\right)$	$\displaystyle=e^{-\nu\pi i}{H^{(2)}_{\nu}}\left(z\right).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.6 Referenced by: §10.47(ii), §10.7(i) Permalink: http://dlmf.nist.gov/10.4.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.4 and Ch.10

10.4.7		${H^{(1)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{% \nu}\left(z\right)-J_{-\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{-\nu\pi i}Y_{\nu}\left(z\right)\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.4, §10.8 Permalink: http://dlmf.nist.gov/10.4.E7 Encodings: TeX, pMML, png See also: Annotations for §10.4 and Ch.10

10.4.8		${H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z% \right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.4, §10.8 Permalink: http://dlmf.nist.gov/10.4.E8 Encodings: TeX, pMML, png See also: Annotations for §10.4 and Ch.10

In (10.4.5), (10.4.7), and (10.4.8) limiting values are taken when $\nu=n$ ; compare (10.2.3) and (10.2.4).