§10.15 Derivatives with Respect to Order

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Keywords:: Bessel functions, 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 $x^{2}(\ifrac{{\mathrm{d}}^{2}W}{{\mathrm{d}x}^{2}})+x(\ifrac{\mathrm{d}W}{% \mathrm{d}x})+(x^{2}-\frac{1}{4})W=\sqrt{2/(\pi x)}\sin x$ , obtained by differentiating (10.2.1) with respect to $\nu$ , setting $\nu=\frac{1}{2}$ , and referring to (10.16.1) for $w$ . This inhomogeneous equation for $W$ can be solved by variation of parameters (§1.13(ii)), using the fact that independent solutions of the corresponding homogeneous equation are $J_{\frac{1}{2}}\left(x\right)$ and $Y_{\frac{1}{2}}\left(x\right)$ 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).
Referenced by:: Erratum (V1.0.1) for Clarifications
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).
See also:: Annotations for Ch.10

Noninteger Values of $\nu$

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See also:: Annotations for §10.15 and Ch.10

10.15.1		$\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.64 Referenced by: §10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1) Permalink: http://dlmf.nist.gov/10.15.E1 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This equation has been generalized to include the additional case of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$ because it will help the user who wants to combine it with (10.15.2). See also: Annotations for §10.15, §10.15 and Ch.10

10.15.2		$\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.65 Referenced by: (10.15.1), §10.15 Permalink: http://dlmf.nist.gov/10.15.E2 Encodings: TeX, pMML, png See also: Annotations for §10.15, §10.15 and Ch.10

Integer Values of $\nu$

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See also:: Annotations for §10.15 and Ch.10

10.15.3		$\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right\|_{\nu=n}=\frac{% \pi}{2}Y_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : integer, $k$ : nonnegative integer, $z$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10

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

10.15.4	$\displaystyle\left.\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}\right\|_{% \nu=n}$	$\displaystyle=-\frac{\pi}{2}J_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}% }\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}Y_{k}\left(z\right)}{k!(n-k)},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $n$ : integer, $k$ : nonnegative integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.67 Permalink: http://dlmf.nist.gov/10.15.E4 Encodings: TeX, pMML, png See also: Annotations for §10.15, §10.15 and Ch.10
10.15.5	$\displaystyle\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right\|_{% \nu=0}$	$\displaystyle=\frac{\pi}{2}Y_{0}\left(z\right),\quad\left.\frac{\partial Y_{% \nu}\left(z\right)}{\partial\nu}\right\|_{\nu=0}=-\frac{\pi}{2}J_{0}\left(z% \right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $z$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10

Half-Integer Values of $\nu$

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Keywords:: Bessel functions, derivatives, with respect to order
See also:: Annotations for §10.15 and Ch.10

For the notations $\operatorname{Ci}$ and $\operatorname{Si}$ see §6.2(ii). When $x>0$ ,

10.15.6	$\displaystyle\left.\frac{\partial J_{\nu}\left(x\right)}{\partial\nu}\right\|_{% \nu=\frac{1}{2}}$	$\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)\sin x% -\operatorname{Si}\left(2x\right)\cos x\right),$
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10
10.15.7	$\displaystyle\left.\frac{\partial J_{\nu}\left(x\right)}{\partial\nu}\right\|_{% \nu=-\frac{1}{2}}$	$\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)\cos x% +\operatorname{Si}\left(2x\right)\sin x\right),$
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10
10.15.8	$\displaystyle\left.\frac{\partial Y_{\nu}\left(x\right)}{\partial\nu}\right\|_{% \nu=\frac{1}{2}}$	$\displaystyle=\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)\cos x% +\left(\operatorname{Si}\left(2x\right)-\pi\right)\sin x\right),$
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10
10.15.9	$\displaystyle\left.\frac{\partial Y_{\nu}\left(x\right)}{\partial\nu}\right\|_{% \nu=-\frac{1}{2}}$	$\displaystyle=-\sqrt{\frac{2}{\pi x}}\left(\operatorname{Ci}\left(2x\right)% \sin x-\left(\operatorname{Si}\left(2x\right)-\pi\right)\cos x\right).$
ⓘ 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, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : 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 See also: Annotations for §10.15, §10.15 and Ch.10

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

§10.15 Derivatives with Respect to Order

Noninteger Values of ν

Integer Values of ν

Half-Integer Values of ν

Noninteger Values of $\nu$

Integer Values of $\nu$

Half-Integer Values of $\nu$