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

§10.38 Derivatives with Respect to Order

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).
Keywords:: modified Bessel functions
Referenced by:: Other Changes
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).
Reported 2010-03-02

10.38.1 $\frac{\partial\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{\ln\/}\nolimits\!\left(% \tfrac{1}{2}z\right)-(\tfrac{1}{2}z)^{\nu}\sum_{{k=0}}^{\infty}\frac{\mathop{% \psi\/}\nolimits\!\left(\nu+k+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+% k+1\right)}\frac{(\frac{1}{4}z^{2})^{k}}{k!},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : 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:: 9.6.42
Referenced by:: ¶ ‣ §10.38, §10.38
Permalink:: http://dlmf.nist.gov/10.38.E1
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10.38.2 $\frac{\partial\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}=% \tfrac{1}{2}\pi\mathop{\csc\/}\nolimits\!\left(\nu\pi\right)\*\left(\frac{% \partial\mathop{I_{{-\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}-\frac{% \partial\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}\right)-\pi% \mathop{\cot\/}\nolimits\!\left(\nu\pi\right)\mathop{K_{{\nu}}\/}\nolimits\!% \left(z\right),$ $\nu\notin\Integer$ .

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\Integer$ : set of all integers, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\notin$ : not an element of, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.43
Referenced by:: ¶ ‣ §10.38, §10.38
Permalink:: http://dlmf.nist.gov/10.38.E2
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¶ Integer Values of $\nu$

10.38.3 $\left.(-1)^{n}\frac{\partial\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)}{% \partial\nu}\right|_{{\nu=n}}=-\mathop{K_{{n}}\/}\nolimits\!\left(z\right)+% \frac{n!}{2(\frac{1}{2}z)^{n}}\sum_{{k=0}}^{{n-1}}(-1)^{k}\frac{(\frac{1}{2}z)% ^{k}\mathop{I_{{k}}\/}\nolimits\!\left(z\right)}{k!(n-k)},$

Symbols:: : : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.44
Referenced by:: §10.38
Permalink:: http://dlmf.nist.gov/10.38.E3
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For $\ifrac{\partial\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4).

10.38.4 $\left.\frac{\partial\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=n}}=\frac{n!}{2(\frac{1}{2}z)^{n}}\sum_{{k=0}}^{{n-1}}\frac{(% \frac{1}{2}z)^{k}\mathop{K_{{k}}\/}\nolimits\!\left(z\right)}{k!(n-k)}.$

Symbols:: : : factorial, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : integer, : nonnegative integer, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.45
Referenced by:: ¶ ‣ §10.38
Permalink:: http://dlmf.nist.gov/10.38.E4
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10.38.5

$\left.\frac{\partial\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=0}}=-\mathop{K_{{0}}\/}\nolimits\!\left(z\right),$

$\left.\frac{\partial\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)}{\partial\nu% }\right|_{{\nu=0}}=0.$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.46
Referenced by:: §10.38
Permalink:: http://dlmf.nist.gov/10.38.E5
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¶ Half-Integer Values of $\nu$

For the notations $\mathop{E_{1}\/}\nolimits$ and $\mathop{\mathrm{Ei}\/}\nolimits$ see §6.2(i). When ,

10.38.6 $\left.\frac{\partial\mathop{I_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=\pm\frac{1}{2}}}=-\frac{1}{\sqrt{2\pi x}}\left(\mathop{E_{1}\/}% \nolimits\!\left(2x\right)e^{x}\pm\mathop{\mathrm{Ei}\/}\nolimits\!\left(2x% \right)e^{{-x}}\right),$

Symbols:: : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : 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
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10.38.7 $\left.\frac{\partial\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)}{\partial\nu% }\right|_{{\nu=\pm\frac{1}{2}}}=\pm\sqrt{\frac{\pi}{2x}}\mathop{E_{1}\/}% \nolimits\!\left(2x\right)e^{x}.$

Symbols:: : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : 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
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For further results see Brychkov and Geddes (2005).

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