Bessel functions

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11: 10.48 Graphs

§10.48 Graphs

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Figure 10.48.7: ${\mathsf{i}}_5^{(1)}\left(x\right)$, ${\mathsf{i}}_5^{(2)}\left(x\right)$, ${\mathsf{k}}_5\left(x\right)$, $0\le x\le 8$. Magnify

12: 10.58 Zeros

§10.58 Zeros

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13: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, or ${\mathsf{h}}_n^{(2)}\left(z\right)$. … ►

$$n{f}_{n-1}(z)-(n+1)f_{n+1}(z)=(2n+1)f_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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$$f_n^{\prime }(z)=-f_{n+1}(z)+(n/z)f_n(z),$$ $n=0,1,\mathrm{\dots }.$

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$$n{g}_{n-1}(z)+(n+1)g_{n+1}(z)=(2n+1)g_n^{\prime }(z),$$ $n=1,2,\mathrm{\dots }$,

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14: 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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15: 10.55 Continued Fractions

§10.55 Continued Fractions

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16: 30.10 Series and Integrals

… ►For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

17: 10.39 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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Generalized Hypergeometric Functions and Hypergeometric Function

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18: 10.47 Definitions and Basic Properties

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Equation (10.47.1)

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Equation (10.47.2)

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§10.47(iv) Interrelations

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19: 10.18 Modulus and Phase Functions

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§10.18(i) Definitions

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§10.18(ii) Basic Properties

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10.18.12 $$M_{\nu }\left(x\right)N_{\nu }\left(x\right)\sin \left({\varphi }_{\nu }\left(x\right)-{\theta }_{\nu }\left(x\right)\right)=\frac{2}{\pi x}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $\sin z$: sine function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.2.24

Permalink:

http://dlmf.nist.gov/10.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(ii), §10.18 and Ch.10

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§10.18(iii) Asymptotic Expansions for Large Argument

… ►The remainder after $k$ terms in (10.18.17) does not exceed the $(k+1)$th term in absolute value and is of the same sign, provided that $k>\nu -\frac{1}{2}$.

20: 10.2 Definitions

§10.2 Definitions

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Bessel Function of the First Kind

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Bessel Function of the Second Kind (Weber’s Function)

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Bessel Functions of the Third Kind (Hankel Functions)

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Branch Conventions

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