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

§10.4 Connection Formulas

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

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

10.4.1

$\mathop{J_{{-n}}\/}\nolimits\!\left(z\right)=(-1)^{n}\mathop{J_{{n}}\/}% \nolimits\!\left(z\right),$

$\mathop{Y_{{-n}}\/}\nolimits\!\left(z\right)=(-1)^{n}\mathop{Y_{{n}}\/}% \nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, : integer and : 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

10.4.2

$\mathop{{H^{{(1)}}_{{-n}}}\/}\nolimits\!\left(z\right)=(-1)^{n}\mathop{{H^{{(1% )}}_{{n}}}\/}\nolimits\!\left(z\right),$

$\mathop{{H^{{(2)}}_{{-n}}}\/}\nolimits\!\left(z\right)=(-1)^{n}\mathop{{H^{{(2% )}}_{{n}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.4.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

10.4.3

$\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\mathop{J_{{\nu}}\/}% \nolimits\!\left(z\right)+i\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right),$

$\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\mathop{J_{{\nu}}\/}% \nolimits\!\left(z\right)-i\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right),$

10.4.4

$\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2}\left(\mathop{{H^{{(1% )}}_{{\nu}}}\/}\nolimits\!\left(z\right)+\mathop{{H^{{(2)}}_{{\nu}}}\/}% \nolimits\!\left(z\right)\right),$

$\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2i}\left(\mathop{{H^{{(% 1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)-\mathop{{H^{{(2)}}_{{\nu}}}\/}% \nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : 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

10.4.5 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\mathop{\csc\/}\nolimits(\nu\pi)% \left(\mathop{Y_{{-\nu}}\/}\nolimits\!\left(z\right)-\mathop{Y_{{\nu}}\/}% \nolimits\!\left(z\right)\mathop{\cos\/}\nolimits(\nu\pi)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.4
Permalink:: http://dlmf.nist.gov/10.4.E5
Encodings:: TeX, pMML, png

10.4.6

$\mathop{{H^{{(1)}}_{{-\nu}}}\/}\nolimits\!\left(z\right)=e^{{\nu\pi i}}\mathop% {{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right),$

$\mathop{{H^{{(2)}}_{{-\nu}}}\/}\nolimits\!\left(z\right)=e^{{-\nu\pi i}}% \mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : base of exponential function, : 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

10.4.7 $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=i\mathop{\csc\/}% \nolimits(\nu\pi)\left(e^{{-\nu\pi i}}\mathop{J_{{\nu}}\/}\nolimits\!\left(z% \right)-\mathop{J_{{-\nu}}\/}\nolimits\!\left(z\right)\right)=\mathop{\csc\/}% \nolimits(\nu\pi)\left(\mathop{Y_{{-\nu}}\/}\nolimits\!\left(z\right)-e^{{-\nu% \pi i}}\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\csc\/}\nolimits z$ : cosecant function, : base of exponential function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.4, §10.8
Permalink:: http://dlmf.nist.gov/10.4.E7
Encodings:: TeX, pMML, png

10.4.8 $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=i\mathop{\csc\/}% \nolimits(\nu\pi)\left(\mathop{J_{{-\nu}}\/}\nolimits\!\left(z\right)-e^{{\nu% \pi i}}\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\right)=\mathop{\csc\/}% \nolimits(\nu\pi)\left(\mathop{Y_{{-\nu}}\/}\nolimits\!\left(z\right)-e^{{\nu% \pi i}}\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\csc\/}\nolimits z$ : cosecant function, : base of exponential function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.4, §10.8
Permalink:: http://dlmf.nist.gov/10.4.E8
Encodings:: TeX, pMML, png

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).