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

§10.27 Connection Formulas

ⓘ

Keywords:: connection formulas, modified Bessel functions
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 $m=1$ .
Permalink:: http://dlmf.nist.gov/10.27
See also:: Annotations for Ch.10

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

10.27.1		$I_{-n}\left(z\right)=I_{n}\left(z\right),$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 and Ch.10

10.27.2		$I_{-\nu}\left(z\right)=I_{\nu}\left(z\right)+(2/\pi)\sin\left(\nu\pi\right)K_{% \nu}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.2 Permalink: http://dlmf.nist.gov/10.27.E2 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.3		$K_{-\nu}\left(z\right)=K_{\nu}\left(z\right).$
ⓘ Symbols: $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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 and Ch.10

10.27.4		$K_{\nu}\left(z\right)=\tfrac{1}{2}\pi\frac{I_{-\nu}\left(z\right)-I_{\nu}\left% (z\right)}{\sin\left(\nu\pi\right)}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $z$ : 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 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

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

10.27.5		$K_{n}\left(z\right)=\frac{(-1)^{n-1}}{2}\*\left(\left.\frac{\partial I_{\nu}% \left(z\right)}{\partial\nu}\right\|_{\nu=n}+\left.\frac{\partial I_{\nu}\left(% z\right)}{\partial\nu}\right\|_{\nu=-n}\right),$
$n=0,\pm 1,\pm 2,\dotsc$ .
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : integer, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.22(iv) Permalink: http://dlmf.nist.gov/10.27.E5 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

In terms of the solutions of (10.2.1),

10.27.6		$I_{\nu}\left(z\right)=e^{\mp\nu\pi i/2}J_{\nu}\left(ze^{\pm\pi i/2}\right),$
$-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : phase, $z$ : 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 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.7		$I_{\nu}\left(z\right)=\tfrac{1}{2}e^{\mp\nu\pi i/2}\left({H^{(1)}_{\nu}}\left(% ze^{\pm\pi i/2}\right)+{H^{(2)}_{\nu}}\left(ze^{\pm\pi i/2}\right)\right),$
$-\pi\leq\pm\operatorname{ph}z\leq\tfrac{1}{2}\pi$ .
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.27.E7 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.8		$K_{\nu}\left(z\right)=\begin{cases}\tfrac{1}{2}\pi ie^{\nu\pi i/2}{H^{(1)}_{% \nu}}\left(ze^{\pi i/2}\right),&-\pi\leq\operatorname{ph}z\leq\tfrac{1}{2}\pi,% \\ -\tfrac{1}{2}\pi ie^{-\nu\pi i/2}{H^{(2)}_{\nu}}\left(ze^{-\pi i/2}\right),&-% \tfrac{1}{2}\pi\leq\operatorname{ph}z\leq\pi.\end{cases}$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.4 Referenced by: §10.16, §10.26(ii), §10.27, §10.27, §10.32(ii), §10.39, §10.40(ii), §10.40(iii), §10.41(i), §10.42, §10.43(ii), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), (9.7.16), §9.7(iii) Permalink: http://dlmf.nist.gov/10.27.E8 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.9		$\pi iJ_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)-e^% {\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.67(i) Permalink: http://dlmf.nist.gov/10.27.E9 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.10		$-\pi Y_{\nu}\left(z\right)=e^{-\nu\pi i/2}K_{\nu}\left(ze^{-\pi i/2}\right)+e^% {\nu\pi i/2}K_{\nu}\left(ze^{\pi i/2}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{1}{2}\pi$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.43(iv) Permalink: http://dlmf.nist.gov/10.27.E10 Encodings: TeX, pMML, png See also: Annotations for §10.27 and Ch.10

10.27.11		$Y_{\nu}\left(z\right)=e^{\pm(\nu+1)\pi i/2}I_{\nu}\left(ze^{\mp\pi i/2}\right)% -(2/\pi)e^{\mp\nu\pi i/2}K_{\nu}\left(ze^{\mp\pi i/2}\right),$
$-\tfrac{1}{2}\pi\leq\pm\operatorname{ph}z\leq\pi$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : 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 See also: Annotations for §10.27 and Ch.10

See also §10.34.

Many properties of modified Bessel functions follow immediately from those of ordinary Bessel functions by application of (10.27.6)–(10.27.8).

10.26 Graphics 10.28 Wronskians and Cross-Products

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