# for modified Bessel functions

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##### 3: 10.44 Sums
###### §10.44(iv) Compendia
They are analogous to the addition theorems for Bessel functions10.23(ii)) and modified Bessel functions10.44(ii)). …
##### 5: 10.74 Methods of Computation
In the case of the modified Bessel function $K_{\nu}\left(z\right)$ see especially Temme (1975). … For applications of generalized Gauss–Laguerre quadrature (§3.5(v)) to the evaluation of the modified Bessel functions $K_{\nu}\left(z\right)$ for $0<\nu<1$ and $0 see Gautschi (2002a). …
##### 6: 10.28 Wronskians and Cross-Products
###### §10.28 Wronskians and Cross-Products
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 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 7: 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. …
##### 8: 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).