10 Bessel Functions Bessel and Hankel Functions 10.10 Continued Fractions 10.12 Generating Function and Associated Series

§10.11 Analytic Continuation

Notes:: For (10.11.1)–(10.11.5) use (10.2.2), (10.2.3), (10.4.3). For (10.11.6)–(10.11.8) take limits. For (10.11.9) use the Schwarz Reflection Principle (§1.10(ii)).
Keywords:: Bessel functions, Hankel functions, analytic continuation
Referenced by:: §10.4
Permalink:: http://dlmf.nist.gov/10.11

When $m\in\Integer$ ,

10.11.1 $\mathop{J_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{m\nu\pi i}}% \mathop{J_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.35
Referenced by:: §10.11, §10.17(i), §10.47(v)
Permalink:: http://dlmf.nist.gov/10.11.E1
Encodings:: TeX, pMML, png

10.11.2 $\mathop{Y_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{-m\nu\pi i}}% \mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)+2i\mathop{\sin\/}\nolimits\!% \left(m\nu\pi\right)\mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\mathop{J_{{% \nu}}\/}\nolimits\!\left(z\right).$

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, $\mathop{\cot\/}\nolimits z$ : cotangent function, : base of exponential function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.36
Referenced by:: §10.11, §10.47(v)
Permalink:: http://dlmf.nist.gov/10.11.E2
Encodings:: TeX, pMML, png

10.11.3 $\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}% \nolimits\!\left(ze^{{m\pi i}}\right)=-\mathop{\sin\/}\nolimits\!\left((m-1)% \nu\pi\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)-e^{{-\nu% \pi i}}\mathop{\sin\/}\nolimits\!\left(m\nu\pi\right)\mathop{{H^{{(2)}}_{{\nu}% }}\/}\nolimits\!\left(z\right),$

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, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.37
Referenced by:: §10.17(i)
Permalink:: http://dlmf.nist.gov/10.11.E3
Encodings:: TeX, pMML, png

10.11.4 $\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{{H^{{(2)}}_{{\nu}}}\/}% \nolimits\!\left(ze^{{m\pi i}}\right)=e^{{\nu\pi i}}\mathop{\sin\/}\nolimits\!% \left(m\nu\pi\right)\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)+% \mathop{\sin\/}\nolimits\!\left((m+1)\nu\pi\right)\mathop{{H^{{(2)}}_{{\nu}}}% \/}\nolimits\!\left(z\right).$

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, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.38
Referenced by:: §10.11, §10.17(i)
Permalink:: http://dlmf.nist.gov/10.11.E4
Encodings:: TeX, pMML, png

10.11.5

$\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(ze^{{\pi i}}\right)=-e^{{-\nu% \pi i}}\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right),$

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

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:: 9.1.39
Referenced by:: §10.11, §10.27
Permalink:: http://dlmf.nist.gov/10.11.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

If $\nu=n$ $(\in\Integer)$ , then limiting values are taken in (10.11.2)–(10.11.4):

10.11.6 $\mathop{Y_{{n}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=(-1)^{{mn}}(\mathop{Y_% {{n}}\/}\nolimits\!\left(z\right)+2im\mathop{J_{{n}}\/}\nolimits\!\left(z% \right)),$

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, : base of exponential function, : integer, : integer and : complex variable
Referenced by:: §10.11
Permalink:: http://dlmf.nist.gov/10.11.E6
Encodings:: TeX, pMML, png

10.11.7 $\mathop{{H^{{(1)}}_{{n}}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=(-1)^{{mn-1}% }((m-1)\mathop{{H^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)+m\mathop{{H^{{(2)% }}_{{n}}}\/}\nolimits\!\left(z\right)),$

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), : base of exponential function, : integer, : integer and : complex variable
Referenced by:: §10.17(i)
Permalink:: http://dlmf.nist.gov/10.11.E7
Encodings:: TeX, pMML, png

10.11.8 $\mathop{{H^{{(2)}}_{{n}}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=(-1)^{{mn}}(% m\mathop{{H^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)+(m+1)\mathop{{H^{{(2)}}% _{{n}}}\/}\nolimits\!\left(z\right)).$

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), : base of exponential function, : integer, : integer and : complex variable
Referenced by:: §10.11, §10.17(i)
Permalink:: http://dlmf.nist.gov/10.11.E8
Encodings:: TeX, pMML, png

For real $\nu$ ,

10.11.9

$\displaystyle\mathop{J_{{\nu}}\/}\nolimits\!\left(\overline{z}\right)=% \overline{\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)},$ $\displaystyle\mathop{Y_{{\nu}}\/}\nolimits\!\left(\overline{z}\right)=% \overline{\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)},$

$\displaystyle\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(\overline{z}\right% )=\overline{\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)},$ $\displaystyle\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(\overline{z}\right% )=\overline{\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)}.$

For complex $\nu$ replace $\nu$ by $\overline{\nu}$ on the right-hand sides.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.10 Continued Fractions 10.12 Generating Function and Associated Series