# Bessel functions and modified Bessel functions

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##### 2: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
10.46.2 $I_{\nu}\left(z\right)=\left(\tfrac{1}{2}z\right)^{\nu}\phi\left(1,\nu+1;\tfrac% {1}{4}z^{2}\right).$
For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
##### 4: 10.73 Physical Applications
###### §10.73(i) Bessel and ModifiedBesselFunctions
Consequently, Bessel functions $J_{n}\left(x\right)$, and modified Bessel functions $I_{n}\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. … … On separation of variables into cylindrical coordinates, the Bessel functions $J_{n}\left(x\right)$, and modified Bessel functions $I_{n}\left(x\right)$ and $K_{n}\left(x\right)$, all appear. …
##### 6: 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).
##### 7: 10.74 Methods of Computation
In the case of the modified Bessel function $K_{\nu}\left(z\right)$ see especially Temme (1975). …
##### 8: 10.44 Sums
###### §10.44(i) Multiplication Theorem
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
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. … If $f(z)$ has a double zero $z_{0}$, or more generally $z_{0}$ is a zero of order $m$, $m=2,3,4,\dotsc$, then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. …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 $\alpha$). …