§10.27 Connection Formulas

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: $I_{\nu }\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: 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: $K_{\nu }\left(z\right)$: 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		$$K_{\nu }\left(z\right)=\frac{1}{2}\pi \frac{I_{-\nu }\left(z\right)-I_{\nu }\left(z\right)}{\sin \left(\nu \pi \right)}.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: 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		$$K_n\left(z\right)=\frac{{(-1)}^{n-1}}{2}\left({\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}+{\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =-n}\right),$$
$n=0,\pm 1,\pm 2,\mathrm{\dots }$.
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$$I_{\nu }\left(z\right)={\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{J}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right),$$
$-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$,
ⓘ Symbols: $J_{\nu }\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: 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		$$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$
$-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: 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
ⓘ Symbols: $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: 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		$$\pi \mathrm{i}{J}_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: 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		$$-\pi Y_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$
$|\mathrm{ph}z|\le \frac{1}{2}\pi$.
ⓘ Symbols: $Y_{\nu }\left(z\right)$: 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: 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		$$Y_{\nu }\left(z\right)={\mathrm{e}}^{\pm (\nu +1)\pi \mathrm{i}/2}{I}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right)-(2/\pi ){\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right),$$
$-\frac{1}{2}\pi \le \pm \mathrm{ph}z\le \pi$.
ⓘ Symbols: $Y_{\nu }\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: 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).