modified Bessel equation

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11: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

12: 10.38 Derivatives with Respect to Order

… ►

10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: 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:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

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13: 10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Graf’s and Gegenbauer’s Addition Theorems

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§10.44(iii) Neumann-Type Expansions

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§10.44(iv) Compendia

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14: 10.47 Definitions and Basic Properties

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Equation (10.47.2)

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15: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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Products

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§10.40(iv) Exponentially-Improved Expansions

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16: 18.34 Bessel Polynomials

§18.34 Bessel Polynomials

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►

§18.34(ii) Orthogonality

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§18.34(iii) Other Properties

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17: Errata

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

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18.34.2

$y_n(x)=y_n(x;2)=2{\pi }^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{𝗄}_n\left(x^{-1}\right),$

${\theta }_n(x)=x^n{y}_n(x^{-1})=2{\pi }^{-1}{x}^{n+1}{\mathrm{e}}^x{𝗄}_n\left(x\right)$

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

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These equations have been generalized to include the additional cases of $\partial J_{-\nu }\left(z\right)/\partial \nu$, $\partial I_{-\nu }\left(z\right)/\partial \nu$, respectively.

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18: 10.27 Connection Formulas

§10.27 Connection Formulas

►Other solutions of (10.25.1) are $I_{-\nu }\left(z\right)$ and $K_{-\nu }\left(z\right)$. ►

10.27.1 $$I_{-n}\left(z\right)=I_n\left(z\right),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

9.6.6

Referenced by:

§10.27, §10.31, §10.44(ii)

Permalink:

http://dlmf.nist.gov/10.27.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.3 $$K_{-\nu }\left(z\right)=K_{\nu }\left(z\right).$$

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Symbols:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.6

Referenced by:

§10.25(iii), §10.30(i), §10.31, §10.47(ii)

Permalink:

http://dlmf.nist.gov/10.27.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

… ►Many properties of modified Bessel functions follow immediately from those of ordinary Bessel functions by application of (10.27.6)–(10.27.8).

19: 10.35 Generating Function and Associated Series

§10.35 Generating Function and Associated Series

… ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, … ►

10.35.4 $$1=I_0\left(z\right)-2I_2\left(z\right)+2I_4\left(z\right)-2I_6\left(z\right)+\mathrm{\cdots },$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable

A&S Ref:

9.6.36

Permalink:

http://dlmf.nist.gov/10.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.35 and Ch.10

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$$\cosh z=I_0\left(z\right)+2I_2\left(z\right)+2I_4\left(z\right)+2I_6\left(z\right)+\mathrm{\dots },$$

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$$\sinh z=2I_1\left(z\right)+2I_3\left(z\right)+2I_5\left(z\right)+\mathrm{\dots }.$$

20: 10.39 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

… ►Principal values on each side of these equations correspond. … ►

Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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