10 Bessel Functions Bessel and Hankel Functions 10.5 Wronskians and Cross-Products 10.7 Limiting Forms

§10.6 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/10.6

§10.6(i) Recurrence Relations
§10.6(ii) Derivatives
§10.6(iii) Cross-Products

§10.6(i) Recurrence Relations

Notes:: For (10.6.1) and (10.6.2) see Olver (1997b, pp. 58–59, 240–242) or Watson (1944, pp. 45, 66, 73–74). (10.6.3) are special cases, and (10.6.4), (10.6.5) follow by straightforward substitution.
Keywords:: Bessel functions, Hankel functions, cylinder functions
Permalink:: http://dlmf.nist.gov/10.6.i

With $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ defined as in §10.2(ii),

10.6.1

$\mathop{\mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right)+\mathop{\mathscr{C}_% {{\nu+1}}\/}\nolimits\!\left(z\right)=(2\nu/z)\mathop{\mathscr{C}_{{\nu}}\/}% \nolimits\!\left(z\right),$

$\mathop{\mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right)-\mathop{\mathscr{C}_% {{\nu+1}}\/}\nolimits\!\left(z\right)=2{\mathop{\mathscr{C}_{{\nu}}\/}% \nolimits^{{\prime}}}\!\left(z\right).$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.27
Referenced by:: §10.51(i), §10.6(i), §10.6(ii), §10.63(i), §10.74(iv)
Permalink:: http://dlmf.nist.gov/10.6.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

10.6.2

${\mathop{\mathscr{C}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=\mathop{% \mathscr{C}_{{\nu-1}}\/}\nolimits\!\left(z\right)-(\nu/z)\mathop{\mathscr{C}_{% {\nu}}\/}\nolimits\!\left(z\right),$

${\mathop{\mathscr{C}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(z\right)=-\mathop{% \mathscr{C}_{{\nu+1}}\/}\nolimits\!\left(z\right)+(\nu/z)\mathop{\mathscr{C}_{% {\nu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.27
Referenced by:: §10.21(ii), §10.22(i), §10.22(ii), §10.5, §10.51(i), §10.6(i), §10.63(i), §10.63(ii), §10.74(vi)
Permalink:: http://dlmf.nist.gov/10.6.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

10.6.3

$\displaystyle{\mathop{J_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=-\mathop% {J_{{1}}\/}\nolimits\!\left(z\right),$ $\displaystyle{\mathop{Y_{{0}}\/}\nolimits^{{\prime}}}\!\left(z\right)=-\mathop% {Y_{{1}}\/}\nolimits\!\left(z\right),$

$\displaystyle{\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits^{{\prime}}}\!\left(z\right% )=-\mathop{{H^{{(1)}}_{{1}}}\/}\nolimits\!\left(z\right),$ $\displaystyle{\mathop{{H^{{(2)}}_{{0}}}\/}\nolimits^{{\prime}}}\!\left(z\right% )=-\mathop{{H^{{(2)}}_{{1}}}\/}\nolimits\!\left(z\right).$

If $f_{\nu}(z)=z^{p}\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(\lambda z^{q}\right)$ , where , and $\lambda$ ( $\neq 0$ ) are real or complex constants, then

10.6.4

$f_{{\nu-1}}(z)+f_{{\nu+1}}(z)=(2\nu/\lambda)z^{{-q}}f_{\nu}(z),$

$(p+\nu q)f_{{\nu-1}}(z)+(p-\nu q)f_{{\nu+1}}(z)=(2\nu/\lambda)z^{{1-q}}f_{{\nu% }}^{{\prime}}(z).$

Symbols:: : complex variable, $\nu$ : complex parameter and $f_{\nu}(z)$
A&S Ref:: 9.1.29
Referenced by:: §10.6(i)
Permalink:: http://dlmf.nist.gov/10.6.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

10.6.5

$zf_{\nu}^{{\prime}}(z)=\lambda qz^{q}f_{{\nu-1}}(z)+(p-\nu q)f_{{\nu}}(z),$

$zf_{\nu}^{{\prime}}(z)=-\lambda qz^{q}f_{{\nu+1}}(z)+(p+\nu q)f_{\nu}(z).$

Symbols:: : complex variable, $\nu$ : complex parameter and $f_{\nu}(z)$
A&S Ref:: 9.1.29
Referenced by:: §10.6(i)
Permalink:: http://dlmf.nist.gov/10.6.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

§10.6(ii) Derivatives

Notes:: For (10.6.6) see Watson (1944, pp. 46). For (10.6.7) use induction combined with the second of (10.6.1).
Keywords:: Bessel functions, Hankel functions, cylinder functions, derivatives
Permalink:: http://dlmf.nist.gov/10.6.ii

For $k=0,1,2,\ldots$ ,

10.6.6

$\left(\frac{1}{z}\frac{d}{dz}\right)^{k}\left(z^{\nu}\mathop{\mathscr{C}_{{\nu% }}\/}\nolimits\!\left(z\right)\right)=z^{{\nu-k}}\mathop{\mathscr{C}_{{\nu-k}}% \/}\nolimits\!\left(z\right),$

$\left(\frac{1}{z}\frac{d}{dz}\right)^{k}(z^{{-\nu}}\mathop{\mathscr{C}_{{\nu}}% \/}\nolimits\!\left(z\right))=(-1)^{k}z^{{-\nu-k}}\mathop{\mathscr{C}_{{\nu+k}% }\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.30
Referenced by:: §10.6(ii)
Permalink:: http://dlmf.nist.gov/10.6.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

10.6.7 ${\mathop{\mathscr{C}_{{\nu}}\/}\nolimits^{{(k)}}}\!\left(z\right)=\frac{1}{2^{% k}}\sum_{{n=0}}^{k}(-1)^{n}\binom{k}{n}\mathop{\mathscr{C}_{{\nu-k+2n}}\/}% \nolimits\!\left(z\right).$

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function, : integer, : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.31
Referenced by:: §10.6(ii)
Permalink:: http://dlmf.nist.gov/10.6.E7
Encodings:: TeX, pMML, png

§10.6(iii) Cross-Products

Notes:: See Goodwin (1949a).
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.6.iii

Let

10.6.8

$p_{\nu}=\mathop{J_{{\nu}}\/}\nolimits\!\left(a\right)\mathop{Y_{{\nu}}\/}% \nolimits\!\left(b\right)-\mathop{J_{{\nu}}\/}\nolimits\!\left(b\right)\mathop% {Y_{{\nu}}\/}\nolimits\!\left(a\right),$

$q_{\nu}=\mathop{J_{{\nu}}\/}\nolimits\!\left(a\right){\mathop{Y_{{\nu}}\/}% \nolimits^{{\prime}}}\!\left(b\right)-{\mathop{J_{{\nu}}\/}\nolimits^{{\prime}% }}\!\left(b\right)\mathop{Y_{{\nu}}\/}\nolimits\!\left(a\right),$

$r_{\nu}={\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a\right)\mathop{Y_{{% \nu}}\/}\nolimits\!\left(b\right)-\mathop{J_{{\nu}}\/}\nolimits\!\left(b\right% ){\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a\right),$

$s_{\nu}={\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a\right){\mathop{Y_{% {\nu}}\/}\nolimits^{{\prime}}}\!\left(b\right)-{\mathop{J_{{\nu}}\/}\nolimits^% {{\prime}}}\!\left(b\right){\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(a% \right),$