# §10.4 Connection Formulas

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 $\displaystyle J_{-n}\left(z\right)$ $\displaystyle=(-1)^{n}J_{n}\left(z\right),$ $\displaystyle Y_{-n}\left(z\right)$ $\displaystyle=(-1)^{n}Y_{n}\left(z\right),$ ⓘ 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 $\displaystyle{H^{(1)}_{-n}}\left(z\right)$ $\displaystyle=(-1)^{n}{H^{(1)}_{n}}\left(z\right),$ $\displaystyle{H^{(2)}_{-n}}\left(z\right)$ $\displaystyle=(-1)^{n}{H^{(2)}_{n}}\left(z\right).$
 10.4.3 $\displaystyle{H^{(1)}_{\nu}}\left(z\right)$ $\displaystyle=J_{\nu}\left(z\right)+iY_{\nu}\left(z\right),$ $\displaystyle{H^{(2)}_{\nu}}\left(z\right)$ $\displaystyle=J_{\nu}\left(z\right)-iY_{\nu}\left(z\right),$
 10.4.4 $\displaystyle J_{\nu}\left(z\right)$ $\displaystyle=\frac{1}{2}\left({H^{(1)}_{\nu}}\left(z\right)+{H^{(2)}_{\nu}}% \left(z\right)\right),$ $\displaystyle Y_{\nu}\left(z\right)$ $\displaystyle=\frac{1}{2i}\left({H^{(1)}_{\nu}}\left(z\right)-{H^{(2)}_{\nu}}% \left(z\right)\right).$
 10.4.5 $J_{\nu}\left(z\right)=\csc\left(\nu\pi\right)\left(Y_{-\nu}\left(z\right)-Y_{% \nu}\left(z\right)\cos\left(\nu\pi\right)\right).$
 10.4.6 $\displaystyle{H^{(1)}_{-\nu}}\left(z\right)$ $\displaystyle=e^{\nu\pi i}{H^{(1)}_{\nu}}\left(z\right),$ $\displaystyle{H^{(2)}_{-\nu}}\left(z\right)$ $\displaystyle=e^{-\nu\pi i}{H^{(2)}_{\nu}}\left(z\right).$
 10.4.7 ${H^{(1)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(e^{-\nu\pi i}J_{% \nu}\left(z\right)-J_{-\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{-\nu\pi i}Y_{\nu}\left(z\right)\right),$
 10.4.8 ${H^{(2)}_{\nu}}\left(z\right)=i\csc\left(\nu\pi\right)\left(J_{-\nu}\left(z% \right)-e^{\nu\pi i}J_{\nu}\left(z\right)\right)=\csc\left(\nu\pi\right)\left(% Y_{-\nu}\left(z\right)-e^{\nu\pi i}Y_{\nu}\left(z\right)\right).$

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).