# expansions in modified spherical Bessel functions

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##### 3: 10.74 Methods of Computation
In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. …
##### 4: 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. … Abramowitz and Stegun (1964): $j_{n}(z)$, $y_{n}(z)$, $h_{n}^{(1)}(z)$, $h_{n}^{(2)}(z)$, for $\mathsf{j}_{n}\left(z\right)$, $\mathsf{y}_{n}\left(z\right)$, ${\mathsf{h}^{(1)}_{n}}\left(z\right)$, ${\mathsf{h}^{(2)}_{n}}\left(z\right)$, respectively, when $n\geq 0$. … Whittaker and Watson (1927): $K_{\nu}\left(z\right)$ for $\cos\left(\nu\pi\right)K_{\nu}\left(z\right)$. For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 5: 7.6 Series Expansions
###### §7.6(i) Power Series
The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$.
###### §7.6(ii) Expansionsin Series of SphericalBesselFunctions
7.6.8 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left({% \mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^{(1)}_{2n+1}}\left(z^{2}% \right)\right),$
##### 6: 18.15 Asymptotic Approximations
These expansions are in terms of Whittaker functions13.14). …The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. … These expansions are in terms of Bessel functions and modified Bessel functions, respectively. …
###### In Terms of BesselFunctions
For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …
##### 7: 33.9 Expansions in Series of Bessel Functions
###### §33.9(i) SphericalBesselFunctions
where the function $\mathsf{j}$ is as in §10.47(ii), $a_{-1}=0$, $a_{0}=(2\ell+1)!!C_{\ell}\left(\eta\right)$, and …
###### §33.9(ii) BesselFunctions and ModifiedBesselFunctions
In this subsection the functions $J$, $I$, and $K$ are as in §§10.2(ii) and 10.25(ii). …
##### 8: 14.15 Uniform Asymptotic Approximations
Here $I$ and $K$ are the modified Bessel functions10.25(ii)). … For asymptotic expansions and explicit error bounds, see Dunster (2003b). … For the Bessel functions $J$ and $Y$ see §10.2(ii), and for the $\operatorname{env}$ functions associated with $J$ and $Y$ see §2.8(iv). … For convergent series expansions see Dunster (2004). … See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_{n}\left(\cos\theta\right)$ as $n\to\infty$ with $\theta$ fixed. …
##### 9: Bibliography S
• J. Segura, P. Fernández de Córdoba, and Yu. L. Ratis (1997) A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.
• J. Segura (2011) Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.
• H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
• A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.
• S. L. Skorokhodov (1985) On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.
• ##### 10: Bibliography B
• L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1988b) Algorithms for evaluating spherical Bessel functions in the complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 28 (12), pp. 1779–1788, 1918.
• C. B. Balogh (1967) Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.
• A. R. Barnett (1996) The Calculation of Spherical Bessel Functions and Coulomb Functions. In Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions, K. Bartschat and J. Hinze (Eds.), pp. 181–202.
• W. J. Braithwaite (1973) Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.
• K. H. Burrell (1974) Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z) for complex arguments. Comm. ACM 17 (9), pp. 524–526.