10 Bessel Functions Modified Bessel Functions 10.33 Continued Fractions 10.35 Generating Function and Associated Series

§10.34 Analytic Continuation

Notes:: See Watson (1944, p. 80) and Olver (1997b, pp. 253, 381). For (10.34.3) take $m=\pm 1$ in (10.34.2), and combine with (10.34.1).
Keywords:: analytic continuation, modified Bessel functions
Referenced by:: §10.27
Permalink:: http://dlmf.nist.gov/10.34

When $m\in\Integer$ ,

10.34.1 $\mathop{I_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{m\nu\pi i}}% \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.30
Referenced by:: §10.30(ii), §10.34, §10.47(v)
Permalink:: http://dlmf.nist.gov/10.34.E1
Encodings:: TeX, pMML, png

10.34.2 $\mathop{K_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=e^{{-m\nu\pi i}}% \mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)-\pi i\mathop{\sin\/}\nolimits\!% \left(m\nu\pi\right)\mathop{\csc\/}\nolimits\!\left(\nu\pi\right)\mathop{I_{{% \nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.31
Referenced by:: §10.27, §10.34, §10.34
Permalink:: http://dlmf.nist.gov/10.34.E2
Encodings:: TeX, pMML, png

10.34.3 $\mathop{I_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=(i/\pi)\left(\pm e^{% {m\nu\pi i}}\mathop{K_{{\nu}}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)\mp e^{% {(m\mp 1)\nu\pi i}}\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.34, §10.40(i), §10.40(iii), §10.41(iv)
Permalink:: http://dlmf.nist.gov/10.34.E3
Encodings:: TeX, pMML, png

10.34.4 $\mathop{K_{{\nu}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=\mathop{\csc\/}% \nolimits(\nu\pi)\left(\pm\mathop{\sin\/}\nolimits(m\nu\pi)\mathop{K_{{\nu}}\/% }\nolimits\!\left(ze^{{\pm\pi i}}\right)\mp\mathop{\sin\/}\nolimits((m\mp 1)% \nu\pi)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.34, §10.40(i)
Permalink:: http://dlmf.nist.gov/10.34.E4
Encodings:: TeX, pMML, png

If $\nu=n(\in\Integer)$ , then limiting values are taken in (10.34.2) and (10.34.4):

10.34.5 $\mathop{K_{{n}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=(-1)^{{mn}}\mathop{K_{% {n}}\/}\nolimits\!\left(z\right)+(-1)^{{n(m-1)-1}}m\pi i\mathop{I_{{n}}\/}% \nolimits\!\left(z\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.34.E5
Encodings:: TeX, pMML, png

10.34.6 $\mathop{K_{{n}}\/}\nolimits\!\left(ze^{{m\pi i}}\right)=\pm(-1)^{{n(m-1)}}m% \mathop{K_{{n}}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)\mp(-1)^{{nm}}(m\mp 1% )\mathop{K_{{n}}\/}\nolimits\!\left(z\right).$

Symbols:: : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer, : integer and : complex variable
Referenced by:: §10.40(i)
Permalink:: http://dlmf.nist.gov/10.34.E6
Encodings:: TeX, pMML, png

For real $\nu$ ,

10.34.7

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

$\mathop{K_{{\nu}}\/}\nolimits\!\left(\overline{z}\right)=\overline{\mathop{K_{% {\nu}}\/}\nolimits\!\left(z\right)}.$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.32
Permalink:: http://dlmf.nist.gov/10.34.E7
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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