# as Bessel functions

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##### 1: 10.1 Special Notation
The main functions treated in this chapter are the Bessel functions $J_{\nu}\left(z\right)$, $Y_{\nu}\left(z\right)$; Hankel functions ${H^{(1)}_{\nu}}\left(z\right)$, ${H^{(2)}_{\nu}}\left(z\right)$; modified Bessel functions $I_{\nu}\left(z\right)$, $K_{\nu}\left(z\right)$; spherical Bessel functions $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$; modified spherical Bessel functions ${\mathsf{i}^{(1)}_{n}}\left(z\right)$, ${\mathsf{i}^{(2)}_{n}}\left(z\right)$, $\mathsf{k}_{n}\left(z\right)$; Kelvin functions $\operatorname{ber}_{\nu}\left(x\right)$, $\operatorname{bei}_{\nu}\left(x\right)$, $\operatorname{ker}_{\nu}\left(x\right)$, $\operatorname{kei}_{\nu}\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 2: 35.5 Bessel Functions of Matrix Argument
###### §35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
##### 3: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
###### §10.46 Generalized and Incomplete BesselFunctions; Mittag-Leffler Function
The function $\phi\left(\rho,\beta;z\right)$ is defined by … The Laplace transform of $\phi\left(\rho,\beta;z\right)$ can be expressed in terms of the Mittag-Leffler function: … For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
##### 5: 10.73 Physical Applications
###### §10.73(ii) Spherical BesselFunctions
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
##### 8: 10.74 Methods of Computation
Similar observations apply to the computation of modified Bessel functions, spherical Bessel functions, and Kelvin functions. …
##### 9: 4.46 Tables
This handbook also includes lists of references for earlier tables, as do Fletcher et al. (1962) and Lebedev and Fedorova (1960). … (These roots are zeros of the Bessel function $J_{3/2}\left(x\right)$; see §10.21.) …
##### 10: 10.72 Mathematical Applications
###### §10.72(i) Differential Equations with Turning Points
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … The order of the approximating Bessel functions, or modified Bessel functions, is $1/(\lambda+2)$, except in the case when $g(z)$ has a double pole at $z_{0}$. … Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …