§10.16 Relations to Other Functions

Notes:: For (10.16.3), (10.16.4) see Miller (1955, p. 43). For (10.16.5) and (10.16.6) see Olver (1997b, pp. 255, 259) and apply (10.27.8). For (10.16.7) and (10.16.8) apply (13.14.4) and (13.14.5). For (10.16.9) combine (10.2.2) and (16.2.1).
Permalink:: http://dlmf.nist.gov/10.16

¶ Elementary Functions

Keywords:: Bessel functions, Hankel functions, elementary functions

10.16.1

$\mathop{J_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\mathop{Y_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\sin\/}\nolimits z,$

$\mathop{J_{{-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=-\mathop{Y_{{\frac{1}{2}}}\/}\nolimits\!\left(z\right)=\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{\cos\/}\nolimits z,$

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, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 10.1.11 and 10.1.12 (modified)
Referenced by:: §10.15, §10.18(iii), §10.59
Permalink:: http://dlmf.nist.gov/10.16.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

10.16.2

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

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

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), $e$ : base of exponential function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/10.16.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

For these and general results when $\nu$ is half an odd integer see §§10.47(ii) and 10.49(i).

¶ Airy Functions

See §§9.6(i) and 9.6(ii).

¶ Parabolic Cylinder Functions

Keywords:: Bessel functions, parabolic cylinder functions

With the notation of §12.14(i),

10.16.3

$\mathop{J_{{\frac{1}{4}}}\/}\nolimits\!\left(z\right)=-2^{{-\frac{1}{4}}}\pi^{{-\frac{1}{2}}}z^{{-\frac{1}{4}}}\left(\mathop{W\/}\nolimits\!\left(0,2z^{{\frac{1}{2}}}\right)-\mathop{W\/}\nolimits\!\left(0,-2z^{{\frac{1}{2}}}\right)\right),$

$\mathop{J_{{-\frac{1}{4}}}\/}\nolimits\!\left(z\right)=2^{{-\frac{1}{4}}}\pi^{{-\frac{1}{2}}}z^{{-\frac{1}{4}}}\left(\mathop{W\/}\nolimits\!\left(0,2z^{{\frac{1}{2}}}\right)+\mathop{W\/}\nolimits\!\left(0,-2z^{{\frac{1}{2}}}\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{W\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function and $z$ : complex variable
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

10.16.4

$\mathop{J_{{\frac{3}{4}}}\/}\nolimits\!\left(z\right)=-2^{{-\frac{1}{4}}}\pi^{{-\frac{1}{2}}}z^{{-\frac{3}{4}}}\left({\mathop{W\/}\nolimits^{{\prime}}}\!\left(0,2z^{{\frac{1}{2}}}\right)-{\mathop{W\/}\nolimits^{{\prime}}}\!\left(0,-2z^{{\frac{1}{2}}}\right)\right),$

$\mathop{J_{{-\frac{3}{4}}}\/}\nolimits\!\left(z\right)=-2^{{-\frac{1}{4}}}\pi^{{-\frac{1}{2}}}z^{{-\frac{3}{4}}}\left({\mathop{W\/}\nolimits^{{\prime}}}\!\left(0,2z^{{\frac{1}{2}}}\right)+{\mathop{W\/}\nolimits^{{\prime}}}\!\left(0,-2z^{{\frac{1}{2}}}\right)\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{W\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function and $z$ : complex variable
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

Principal values on each side of these equations correspond.

¶ Confluent Hypergeometric Functions

Keywords:: Bessel functions, Hankel functions, confluent hypergeometric functions

10.16.5 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{{\mp iz}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}\mathop{M\/}\nolimits\!\left(\nu+\tfrac{1}{2},2\nu+1,\pm 2iz\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.69 (modified)
Referenced by:: §10.16, §10.39
Permalink:: http://dlmf.nist.gov/10.16.E5
Encodings:: TeX, pMML, png

10.16.6 $\rselection{\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\\ \mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)}=\mp 2\pi^{{-\frac{1}{2}}}ie^{{\mp\nu\pi i}}(2z)^{\nu}\* e^{{\pm iz}}\mathop{U\/}\nolimits(\nu+\tfrac{1}{2},2\nu+1,\mp 2iz).$

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), $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.16, §10.17(v)
Permalink:: http://dlmf.nist.gov/10.16.E6
Encodings:: TeX, pMML, png

For the functions $\mathop{M\/}\nolimits$ and $\mathop{U\/}\nolimits$ see §13.2(i).

10.16.7 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{e^{{\mp(2\nu+1)\pi i/4}}}{2^{{2\nu}}\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}(2z)^{{-\frac{1}{2}}}\mathop{M\/}\nolimits _{{0,\nu}}(\pm 2iz),$ $2\nu\neq-1,-2-3,\ldots$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $e$ : base of exponential function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E7
Encodings:: TeX, pMML, png

10.16.8 $\rselection{\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\\ \mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)}=e^{{\mp(2\nu+1)\pi i/4}}\left(\frac{2}{\pi z}\right)^{{\frac{1}{2}}}\mathop{W\/}\nolimits _{{0,\nu}}(\mp 2iz).$

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), $e$ : base of exponential function, $\mathop{W\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E8
Encodings:: TeX, pMML, png

For the functions $\mathop{M\/}\nolimits _{{0,\nu}}$ and $\mathop{W\/}\nolimits _{{0,\nu}}$ see §13.14(i).

In all cases principal branches correspond at least when $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi$ .

¶ Generalized Hypergeometric Functions

Keywords:: Bessel functions, generalized hypergeometric functions

10.16.9 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)}\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits\!\left(-;\nu+1;-\tfrac{1}{4}z^{2}\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.16
Permalink:: http://dlmf.nist.gov/10.16.E9
Encodings:: TeX, pMML, png

For $\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits$ see (16.2.1).

With $\mathop{\mathbf{F}\/}\nolimits{}{}$ as in §15.2(i), and with $z$ and $\nu$ fixed,

10.16.10 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{\nu}\lim\mathop{{{\mathbf{F}}}\/}\nolimits\!\left(\lambda,\mu;\nu+1;-z^{2}/(4\lambda\mu)\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.70 (modified)
Referenced by:: §10.39
Permalink:: http://dlmf.nist.gov/10.16.E10
Encodings:: TeX, pMML, png

as $\lambda$ and $\mu\to\infty$ in $\Complex$ . For this result see Watson (1944, §5.7).