§10.4 Connection Formulas

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

10.4.1		$J_{-n}\left(z\right)$	$={(-1)}^n{J}_n\left(z\right),$
		$Y_{-n}\left(z\right)$	$={(-1)}^n{Y}_n\left(z\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: 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		$H_{-n}^{(1)}\left(z\right)$	$={(-1)}^n{H}_n^{(1)}\left(z\right),$
		$H_{-n}^{(2)}\left(z\right)$	$={(-1)}^n{H}_n^{(2)}\left(z\right).$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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		$H_{\nu }^{(1)}\left(z\right)$	$=J_{\nu }\left(z\right)+\mathrm{i}{Y}_{\nu }\left(z\right),$
		$H_{\nu }^{(2)}\left(z\right)$	$=J_{\nu }\left(z\right)-\mathrm{i}{Y}_{\nu }\left(z\right),$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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		$J_{\nu }\left(z\right)$	$={\displaystyle \frac{1}{2}}\left(H_{\nu }^{(1)}\left(z\right)+H_{\nu }^{(2)}\left(z\right)\right),$
		$Y_{\nu }\left(z\right)$	$={\displaystyle \frac{1}{2\mathrm{i}}}\left(H_{\nu }^{(1)}\left(z\right)-H_{\nu }^{(2)}\left(z\right)\right).$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: 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		$H_{-\nu }^{(1)}\left(z\right)$	$={\mathrm{e}}^{\nu \pi \mathrm{i}}{H}_{\nu }^{(1)}\left(z\right),$
		$H_{-\nu }^{(2)}\left(z\right)$	$={\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(z\right).$
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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		$$\begin{array}{ll}{H}_{\nu }^{(1)}\left(z\right)& =\mathrm{i}\csc \left(\nu \pi \right)\left({\mathrm{e}}^{-\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)-J_{-\nu }\left(z\right)\right)\\ & =\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right),\end{array}$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: 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		$$H_{\nu }^{(2)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left(J_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right).$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: 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).

See also §10.11.