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21: 10.42 Zeros

§10.42 Zeros

►Properties of the zeros of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ may be deduced from those of $J_{\nu }\left(z\right)$ and $H_{\nu }^{(1)}\left(z\right)$, respectively, by application of the transformations (10.27.6) and (10.27.8). ►For example, if $\nu$ is real, then the zeros of $I_{\nu }\left(z\right)$ are all complex unless for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. … ►For $z$-zeros of $K_{\nu }\left(z\right)$, with complex $\nu$, see Ferreira and Sesma (2008). … ►

22: 10.39 Relations to Other Functions

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Elementary Functions

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10.39.2 $$K_{\frac{1}{2}}\left(z\right)=K_{-\frac{1}{2}}\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/10.39.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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23: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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${\mathrm{D}}_0(\nu ,\mu ,z)={\mathrm{M}}_{\nu }^{(3)}\left(z\right){\mathrm{M}}_{\mu }^{(4)}\left(z\right)-{\mathrm{M}}_{\nu }^{(4)}\left(z\right){\mathrm{M}}_{\mu }^{(3)}\left(z\right),$

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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$${\mathrm{Ds}}_0(n,m,z)={\mathrm{Ms}}_n^{(3)}\left(z\right){\mathrm{Ms}}_m^{(4)}\left(z\right)-{\mathrm{Ms}}_n^{(4)}\left(z\right){\mathrm{Ms}}_m^{(3)}\left(z\right),$$

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§28.28(v) Compendia

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24: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►The upper signs correspond to ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when …and for ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ when …

25: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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26: 11.12 Physical Applications

§11.12 Physical Applications

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27: 10.38 Derivatives with Respect to Order

§10.38 Derivatives with Respect to Order

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10.38.2 $$\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }=\frac{1}{2}\pi \csc \left(\nu \pi \right)\left(\frac{\partial I_{-\nu }\left(z\right)}{\partial \nu }-\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }\right)-\pi \cot \left(\nu \pi \right)K_{\nu }\left(z\right),$$ $\nu \notin \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $\mathbb{Z}$: set of all integers, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.43

Referenced by:

(10.38.1), §10.38, §10.38

Permalink:

http://dlmf.nist.gov/10.38.E2

Encodings:

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See also:

Annotations for §10.38 and Ch.10

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10.38.3 $${{(-1)}^n\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-K_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)},$$

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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:

pMML, png, TeX

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). … ►

${{\displaystyle \frac{\partial I_{\nu }\left(z\right)}{\partial \nu }}|}_{\nu =0}=-K_0\left(z\right),$

…

28: 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:

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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:

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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).

29: 11.3 Graphics

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►►

Figure 11.3.1: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 12$ and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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►►

Figure 11.3.2: ${𝐊}_{\nu }\left(x\right)$ for and $\nu =0,\frac{1}{2},1,\frac{3}{2},2,3$. Magnify

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►►

Figure 11.3.3: ${𝐇}_{\nu }\left(x\right)$ for $0\le x\le 12$ and $\nu =-3,-2,-\frac{3}{2},-1,-\frac{1}{2}$. Magnify

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►►

Figure 11.3.4: ${𝐊}_{\nu }\left(x\right)$ for and $\nu =-4,-3,-2,-1,0$. … Magnify

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§11.3(ii) Modified Struve Functions

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30: 10.30 Limiting Forms

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§10.30(i) $z\to 0$

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10.30.1 $$I_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.7

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

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10.30.3 $$K_0\left(z\right)\sim -\ln z.$$

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Symbols:

$\sim$: asymptotic equality, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.8

Referenced by:

§10.30(i)

Permalink:

http://dlmf.nist.gov/10.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(i), §10.30 and Ch.10

►For $K_{\nu }\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7). … ►For $K_{\nu }\left(z\right)$ see (10.25.3).