10 Bessel Functions Modified Bessel Functions 10.37 Inequalities; Monotonicity 10.39 Relations to Other Functions

§10.38 Derivatives with Respect to Order

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Keywords:: derivatives with respect to order, modified Bessel functions
Notes:: (10.38.1) is obtained by differentiation of (10.25.2); compare (10.15.1). For (10.38.2) use (10.27.4). (10.38.3)–(10.38.5) are proved in a similar way to (10.15.3)–(10.15.5). (10.38.6) and (10.38.7) are stated without proof and in a slightly different notation in Magnus et al. (1966, §3.3.3). Both cases of (10.38.6) can be derived by a method analogous to that used for (10.15.6) and (10.15.7). (10.38.7) follows from (10.38.2) and (10.38.6).
Referenced by:: Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/10.38
Clarification (effective with 1.0.1):: A clarification for negative integer values of $\nu$ was inserted after (10.38.3).
See also:: Annotations for Ch.10

10.38.1		$\frac{\partial I_{\pm\nu}\left(z\right)}{\partial\nu}=\pm I_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}\frac{(% \frac{1}{4}z^{2})^{k}}{k!},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\ln\NVar{z}$ : 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: 9.6.42 Referenced by: §10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1) Permalink: http://dlmf.nist.gov/10.38.E1 Encodings: TeX, pMML, png Addition (effective with 1.0.17): This equation has been generalized to include the additional case of $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$ because it will help the user who wants to combine it with (10.38.2). See also: Annotations for §10.38 and Ch.10

10.38.2		$\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$
$\nu\notin\mathbb{Z}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\cot\NVar{z}$ : cotangent function, $\mathbb{Z}$ : set of all integers, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\notin$ : not an element of, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : 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: 9.6.43 Referenced by: (10.38.1), §10.38, §10.38 Permalink: http://dlmf.nist.gov/10.38.E2 Encodings: TeX, pMML, png See also: Annotations for §10.38 and Ch.10

Integer Values of $\nu$

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See also:: Annotations for §10.38 and Ch.10

10.38.3		$\left.(-1)^{n}\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right\|_{\nu=n% }=-K_{n}\left(z\right)+\frac{n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}(-1)^{k}% \frac{(\frac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-k)},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : 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: 9.6.44 Referenced by: §10.38 Permalink: http://dlmf.nist.gov/10.38.E3 Encodings: TeX, pMML, png See also: Annotations for §10.38, §10.38 and Ch.10

For $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4).

10.38.4		$\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right\|_{\nu=n}=\frac{% n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\frac{1}{2}z)^{k}K_{k}\left(z% \right)}{k!(n-k)}.$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : 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: 9.6.45 Referenced by: §10.38 Permalink: http://dlmf.nist.gov/10.38.E4 Encodings: TeX, pMML, png See also: Annotations for §10.38, §10.38 and Ch.10

10.38.5	$\displaystyle\left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right\|_{% \nu=0}$	$\displaystyle=-K_{0}\left(z\right),$
10.38.5	$\displaystyle\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right\|_{% \nu=0}$	$\displaystyle=0.$
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : 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: 9.6.46 Referenced by: §10.38 Permalink: http://dlmf.nist.gov/10.38.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.38, §10.38 and Ch.10

Half-Integer Values of $\nu$

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See also:: Annotations for §10.38 and Ch.10

For the notations $E_{1}$ and $\operatorname{Ei}$ see §6.2(i). When $x>0$ ,

10.38.6		$\left.\frac{\partial I_{\nu}\left(x\right)}{\partial\nu}\right\|_{\nu=\pm\frac{% 1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_{1}\left(2x\right)e^{x}\pm\operatorname% {Ei}\left(2x\right)e^{-x}\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable and $\nu$ : complex parameter A&S Ref: 10.2.32 and 10.2.33 (both corrected) Referenced by: §10.38, Ch.10 Permalink: http://dlmf.nist.gov/10.38.E6 Encodings: TeX, pMML, png See also: Annotations for §10.38, §10.38 and Ch.10

10.38.7		$\left.\frac{\partial K_{\nu}\left(x\right)}{\partial\nu}\right\|_{\nu=\pm\frac{% 1}{2}}=\pm\sqrt{\frac{\pi}{2x}}E_{1}\left(2x\right)e^{x}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable and $\nu$ : complex parameter A&S Ref: 10.2.34 (corrected) Referenced by: §10.38, Ch.10 Permalink: http://dlmf.nist.gov/10.38.E7 Encodings: TeX, pMML, png See also: Annotations for §10.38, §10.38 and Ch.10

For further results see Brychkov and Geddes (2005).

10.37 Inequalities; Monotonicity 10.39 Relations to Other Functions

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§10.38 Derivatives with Respect to Order

Integer Values of ν

Half-Integer Values of ν

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Integer Values of $\nu$

Half-Integer Values of $\nu$