# The asymptotic expansion of bessel functions of large order

@article{Olver1954TheAE, title={The asymptotic expansion of bessel functions of large order}, author={Frank W. J. Olver}, journal={Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences}, year={1954}, volume={247}, pages={328 - 368} }

New expansions are obtained for the functions Iv{yz), ) and their derivatives in terms of elementary functions, and for the functions J v(vz), Yv{vz), H fvz) and their derivatives in terms of Airy functions, which are uniformly valid with respect to z when | | is large. New series for the zeros and associated values are derived by reversion and used to determine the distribution of the zeros of functions of large order in the z-plane. Particular attention is paid to the complex zeros of 7„(z…

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## References

SHOWING 1-4 OF 4 REFERENCES

The asymptotic solution of linear differential equations of the second order for large values of a parameter

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1954

Asymptotic solutions of the differential equations d2w/dz2 = {uzn +f{z)} w 0, 1) for large positive values of u, have the formal expansions P(z) 1+ Z5=1 (f> l+ ^ M y us s 5=0 IIs where P is an…

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1950

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their…

Uniform asymptotic formulae for functions with transition points

- Mathematics
- 1950

in which R(x, w), Q(x, w) are regular at x=O, w=0 and f(x) has a simple zero at x=0; for (see ?2) this may be reduced to the form (1.1) by a change of variables, regular at x = z = 0. When all the…

Theory o f functions o f a complex variable

- 1944