10 Bessel Functions Modified Bessel Functions 10.26 Graphics 10.28 Wronskians and Cross-Products

§10.27 Connection Formulas

Notes:: For (10.27.1)–(10.27.6) and (10.27.8) see Olver (1997b, pp. 60–61 and 250–252), Watson (1944, pp. 77–79), and (10.11.5). For the other equations combine these results with (10.4.4), and also use (10.34.2) with .
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.27

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:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer and : 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
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10.27.2 $\mathop{I_{{-\nu}}\/}\nolimits\!\left(z\right)=\mathop{I_{{\nu}}\/}\nolimits\!% \left(z\right)+(2/\pi)\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)\mathop{K_{% {\nu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.2
Permalink:: http://dlmf.nist.gov/10.27.E2
Encodings:: TeX, pMML, png

10.27.3 $\mathop{K_{{-\nu}}\/}\nolimits\!\left(z\right)=\mathop{K_{{\nu}}\/}\nolimits\!% \left(z\right).$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : 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
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10.27.4 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}\pi\frac{\mathop{I_{% {-\nu}}\/}\nolimits\!\left(z\right)-\mathop{I_{{\nu}}\/}\nolimits\!\left(z% \right)}{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}.$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.2
Referenced by:: §10.30(i), §10.31, §10.38, §10.45, §10.47(iv), §8.6(i)
Permalink:: http://dlmf.nist.gov/10.27.E4
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When $\nu$ is an integer limiting values are taken:

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

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.22(iv)
Permalink:: http://dlmf.nist.gov/10.27.E5
Encodings:: TeX, pMML, png

In terms of the solutions of (10.2.1),

10.27.6 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=e^{{\mp\nu\pi i/2}}\mathop{J_{{% \nu}}\/}\nolimits\!\left(ze^{{\pm\pi i/2}}\right),$ $-\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.3 (modified)
Referenced by:: §10.26(ii), §10.27, §10.27, §10.31, §10.33, §10.35, §10.39, §10.41(i), §10.42, §10.43(ii), §10.43(iv), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), §10.67(i), §11.4(i)
Permalink:: http://dlmf.nist.gov/10.27.E6
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10.27.7 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{{\mp\nu\pi i/2}}% \left(\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(ze^{{\pm\pi i/2}}\right)+% \mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(ze^{{\pm\pi i/2}}\right)\right),$ $-\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{2}\pi$ .

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, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.27.E7
Encodings:: TeX, pMML, png

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 $\pi i\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=e^{{-\nu\pi i/2}}\mathop{K_% {{\nu}}\/}\nolimits\!\left(ze^{{-\pi i/2}}\right)-e^{{\nu\pi i/2}}\mathop{K_{{% \nu}}\/}\nolimits\!\left(ze^{{\pi i/2}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.67(i)
Permalink:: http://dlmf.nist.gov/10.27.E9
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10.27.10 $-\pi\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=e^{{-\nu\pi i/2}}\mathop{K_{% {\nu}}\/}\nolimits\!\left(ze^{{-\pi i/2}}\right)+e^{{\nu\pi i/2}}\mathop{K_{{% \nu}}\/}\nolimits\!\left(ze^{{\pi i/2}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, : base of exponential function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.43(iv)
Permalink:: http://dlmf.nist.gov/10.27.E10
Encodings:: TeX, pMML, png

10.27.11 $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=e^{{\pm(\nu+1)\pi i/2}}\mathop{I% _{{\nu}}\/}\nolimits\!\left(ze^{{\mp\pi i/2}}\right)-(2/\pi)e^{{\mp\nu\pi i/2}% }\mathop{K_{{\nu}}\/}\nolimits\!\left(ze^{{\mp\pi i/2}}\right),$ $-\tfrac{1}{2}\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.5 (modified)
Referenced by:: §11.6(i)
Permalink:: http://dlmf.nist.gov/10.27.E11
Encodings:: TeX, pMML, png