10 Bessel Functions Bessel and Hankel Functions 10.14 Inequalities; Monotonicity 10.16 Relations to Other Functions

§10.15 Derivatives with Respect to Order

Notes:: For (10.15.1) see Watson (1944, pp. 61–62) or Olver (1997b, p. 243). For (10.15.2) use (10.2.3). For (10.15.3)–(10.15.5) see Olver (1997b, p. 244). (10.15.6)–(10.15.9) appear without proof in Magnus et al. (1966, §3.3.3). To derive (10.15.6) the left-hand side satisfies the differential equation $x^{2}(\ifrac{{d}^{2}W}{{dx}^{2}})+x(\ifrac{dW}{dx})+(x^{2}-\frac{1}{4})W=\sqrt% {2/(\pi x)}\mathop{\sin\/}\nolimits x$ , obtained by differentiating (10.2.1) with respect to $\nu$ , setting $\nu=\frac{1}{2}$ , and referring to (10.16.1) for . This inhomogeneous equation for can be solved by variation of parameters (§1.13(ii)), using the fact that independent solutions of the corresponding homogeneous equation are $\mathop{J_{{\frac{1}{2}}}\/}\nolimits\!\left(x\right)$ and $\mathop{Y_{{\frac{1}{2}}}\/}\nolimits\!\left(x\right)$ with Wronskian $2/(\pi x)$ , and subsequently referring to (6.2.9) and (6.2.11). Similarly for (10.15.7). (10.15.8) and (10.15.9) follow from (10.15.2), (10.15.6), (10.15.7), and (10.16.1).
Keywords:: Bessel functions, derivatives
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/10.15
Clarification (effective with 1.0.1):: A clarification for negative integer values of $\nu$ was inserted after (10.15.3).
Reported 2010-03-02

¶ Noninteger Values of $\nu$

10.15.1 $\frac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{\ln\/}\nolimits\!\left(% \tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{\nu}\sum_{{k=0}}^{\infty}(-1)^{k}\frac{% \mathop{\psi\/}\nolimits\!\left(\nu+k+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\nu+k+1\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.64
Referenced by:: §10.15, §10.38
Permalink:: http://dlmf.nist.gov/10.15.E1
Encodings:: TeX, pMML, png

10.15.2 $\frac{\partial\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\left(\frac{\partial\mathop{J_{{% \nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}-\pi\mathop{Y_{{\nu}}\/}% \nolimits\!\left(z\right)\right)-\mathop{\csc\/}\nolimits\!\left(\nu\pi\right)% \frac{\partial\mathop{J_{{-\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}-\pi% \mathop{J_{{\nu}}\/}\nolimits\!\left(z\right).$

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, $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.65
Referenced by:: §10.15
Permalink:: http://dlmf.nist.gov/10.15.E2
Encodings:: TeX, pMML, png

¶ Integer Values of $\nu$

10.15.3 $\left.\frac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=n}}=\frac{\pi}{2}\mathop{Y_{{n}}\/}\nolimits\!\left(z\right)+% \frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{{k=0}}^{{n-1}}\frac{(\tfrac{1}{2}z)^{k}% \mathop{J_{{k}}\/}\nolimits\!\left(z\right)}{k!(n-k)}.$

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, : : factorial, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.66
Referenced by:: ¶ ‣ §10.15, §10.15, §10.38
Permalink:: http://dlmf.nist.gov/10.15.E3
Encodings:: TeX, pMML, png

For $\ifrac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.2.4) and (10.15.3).

10.15.4 $\left.\frac{\partial\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=n}}=-\frac{\pi}{2}\mathop{J_{{n}}\/}\nolimits\!\left(z\right)+% \frac{n!}{2(\tfrac{1}{2}z)^{{n}}}\sum_{{k=0}}^{{n-1}}\frac{(\tfrac{1}{2}z)^{k}% \mathop{Y_{{k}}\/}\nolimits\!\left(z\right)}{k!(n-k)},$

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, : : factorial, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.67
Permalink:: http://dlmf.nist.gov/10.15.E4
Encodings:: TeX, pMML, png

10.15.5 $\left.\frac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=0}}=\frac{\pi}{2}\mathop{Y_{{0}}\/}\nolimits\!\left(z\right),% \quad\left.\frac{\partial\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)}{% \partial\nu}\right|_{{\nu=0}}=-\frac{\pi}{2}\mathop{J_{{0}}\/}\nolimits\!\left% (z\right).$

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, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.68
Referenced by:: §10.15, §10.38
Permalink:: http://dlmf.nist.gov/10.15.E5
Encodings:: TeX, pMML, png

¶ Half-Integer Values of $\nu$

Keywords:: Bessel functions, derivatives

For the notations $\mathop{\mathrm{Ci}\/}\nolimits$ and $\mathop{\mathrm{Si}\/}\nolimits$ see §6.2(ii). When ,

10.15.6 $\left.\frac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=\frac{1}{2}}}=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/% }\nolimits\!\left(2x\right)\mathop{\sin\/}\nolimits x-\mathop{\mathrm{Si}\/}% \nolimits\!\left(2x\right)\mathop{\cos\/}\nolimits x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, : real variable and $\nu$ : complex parameter
A&S Ref:: 10.1.41 (corrected)
Referenced by:: §10.15, §10.38, Ch.10
Permalink:: http://dlmf.nist.gov/10.15.E6
Encodings:: TeX, pMML, png

10.15.7 $\left.\frac{\partial\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=-\frac{1}{2}}}=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}% \/}\nolimits\!\left(2x\right)\mathop{\cos\/}\nolimits x+\mathop{\mathrm{Si}\/}% \nolimits\!\left(2x\right)\mathop{\sin\/}\nolimits x\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, : real variable and $\nu$ : complex parameter
A&S Ref:: 10.1.42 (corrected)
Referenced by:: §10.15, §10.38
Permalink:: http://dlmf.nist.gov/10.15.E7
Encodings:: TeX, pMML, png

10.15.8 $\left.\frac{\partial\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=\frac{1}{2}}}=\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}\/% }\nolimits\!\left(2x\right)\mathop{\cos\/}\nolimits x+\left(\mathop{\mathrm{Si% }\/}\nolimits\!\left(2x\right)-\pi\right)\mathop{\sin\/}\nolimits x\right),$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, : real variable and $\nu$ : complex parameter
A&S Ref:: 10.1.43 (corrected)
Referenced by:: §10.15
Permalink:: http://dlmf.nist.gov/10.15.E8
Encodings:: TeX, pMML, png

10.15.9 $\left.\frac{\partial\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=-\frac{1}{2}}}=-\sqrt{\frac{2}{\pi x}}\left(\mathop{\mathrm{Ci}% \/}\nolimits\!\left(2x\right)\mathop{\sin\/}\nolimits x-\left(\mathop{\mathrm{% Si}\/}\nolimits\!\left(2x\right)-\pi\right)\mathop{\cos\/}\nolimits x\right).$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral, : real variable and $\nu$ : complex parameter
A&S Ref:: 10.1.44 (corrected)
Referenced by:: §10.15, Ch.10
Permalink:: http://dlmf.nist.gov/10.15.E9
Encodings:: TeX, pMML, png

For further results see Brychkov and Geddes (2005) and Landau (1999, 2000).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.14 Inequalities; Monotonicity 10.16 Relations to Other Functions