§10.38 Derivatives with Respect to Order

10.38.3		$${{(-1)}^n\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-K_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)},$$
ⓘ Symbols: $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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 $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4).

10.38.4		$${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$
ⓘ Symbols: $!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}$	$=-K_0\left(z\right),$
		${{\displaystyle \frac{\partial K_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}$	$=0.$
ⓘ Symbols: $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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

For the notations $E_1$ and $\mathrm{Ei}$ see §6.2(i). When $x>0$,

10.38.6		$${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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		$${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: 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).