§10.27 Connection Formulas

Other solutions of (10.25.1) are $\mathop{I_{-\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{K_{-\nu}\/}\nolimits\!\left(z\right)$.

 10.27.1 $\mathop{I_{-n}\/}\nolimits\!\left(z\right)=\mathop{I_{n}\/}\nolimits\!\left(z% \right),$ Symbols: $\mathop{I_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the first kind, $n$: integer and $z$: complex variable A&S Ref: 9.6.6 Referenced by: §10.27, §10.31, §10.44(ii) Permalink: http://dlmf.nist.gov/10.27.E1 Encodings: TeX, pMML, png See also: Annotations for 10.27
 10.27.2 $\mathop{I_{-\nu}\/}\nolimits\!\left(z\right)=\mathop{I_{\nu}\/}\nolimits\!% \left(z\right)+(2/\pi)\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{K_{% \nu}\/}\nolimits\!\left(z\right),$
 10.27.3 $\mathop{K_{-\nu}\/}\nolimits\!\left(z\right)=\mathop{K_{\nu}\/}\nolimits\!% \left(z\right).$ Symbols: $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.6 Referenced by: §10.25(iii), §10.30(i), §10.31, §10.47(ii) Permalink: http://dlmf.nist.gov/10.27.E3 Encodings: TeX, pMML, png See also: Annotations for 10.27
 10.27.4 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi\frac{\mathop{I_{-% \nu}\/}\nolimits\!\left(z\right)-\mathop{I_{\nu}\/}\nolimits\!\left(z\right)}{% \mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}.$

When $\nu$ is an integer limiting values are taken:

 10.27.5 $\mathop{K_{n}\/}\nolimits\!\left(z\right)=\frac{(-1)^{n-1}}{2}\*\left(\left.% \frac{\partial\mathop{I_{\nu}\/}\nolimits\!\left(z\right)}{\partial\nu}\right|% _{\nu=n}+\left.\frac{\partial\mathop{I_{\nu}\/}\nolimits\!\left(z\right)}{% \partial\nu}\right|_{\nu=-n}\right),$ $n=0,\pm 1,\pm 2,\ldots$.

In terms of the solutions of (10.2.1),

 10.27.6 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)=e^{\mp\nu\pi i/2}\mathop{J_{\nu}\/% }\nolimits\!\left(ze^{\pm\pi i/2}\right),$ $-\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi$,
 10.27.7 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{\mp\nu\pi i/2}\left% (\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(ze^{\pm\pi i/2}\right)+\mathop{{H^% {(2)}_{\nu}}\/}\nolimits\!\left(ze^{\pm\pi i/2}\right)\right),$ $-\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi$.
 10.27.8 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\begin{cases}\tfrac{1}{2}\pi ie^{% \nu\pi i/2}\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(ze^{\pi i/2}\right),&-% \pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi,\\ -\tfrac{1}{2}\pi ie^{-\nu\pi i/2}\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(ze% ^{-\pi i/2}\right),&-\tfrac{1}{2}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq% \pi.\end{cases}$
 10.27.9 $\pi i\mathop{J_{\nu}\/}\nolimits\!\left(z\right)=e^{-\nu\pi i/2}\mathop{K_{\nu% }\/}\nolimits\!\left(ze^{-\pi i/2}\right)-e^{\nu\pi i/2}\mathop{K_{\nu}\/}% \nolimits\!\left(ze^{\pi i/2}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$.
 10.27.10 $-\pi\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)=e^{-\nu\pi i/2}\mathop{K_{\nu}% \/}\nolimits\!\left(ze^{-\pi i/2}\right)+e^{\nu\pi i/2}\mathop{K_{\nu}\/}% \nolimits\!\left(ze^{\pi i/2}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$.
 10.27.11 $\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)=e^{\pm(\nu+1)\pi i/2}\mathop{I_{% \nu}\/}\nolimits\!\left(ze^{\mp\pi i/2}\right)-(2/\pi)e^{\mp\nu\pi i/2}\mathop% {K_{\nu}\/}\nolimits\!\left(ze^{\mp\pi i/2}\right),$ $-\tfrac{1}{2}\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$.