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== History ==
The idea was introduced in 1973 by William J. Lentz<ref name=":0" /> and was simplified by him in 1982.<ref>{{Cite book|last=J.|first=Lentz, W.|url=http://worldcat.org/oclc/227549426|title=A Simplification of Lentz's Algorithm.|date=August 1982|publisher=Defense Technical Information Center|oclc=227549426}}</ref> Lentz suggested that calculating ratios of spherical Bessel functions of complex arguments can be difficult. He developed a new continued fraction technique for calculating the ratios of spherical Bessel functions of consecutive order. This method was an improvement compared to other methods because started from the beginning of the continued fraction rather than the tail, had a built-in check for convergence, and was numerically stable. The original algorithm uses algebra to bypass a zero in either the numerator or denominator. <ref>{{Cite journal|last=Lentz|first=William J.|date=1976-03-01|title=Generating Bessel functions in Mie scattering calculations using continued fractions|url=http://dx.doi.org/10.1364/ao.15.000668|journal=Applied Optics|volume=15|issue=3|pages=668–671|doi=10.1364/ao.15.000668|pmid=20165036 |bibcode=1976ApOpt..15..668L |issn=0003-6935}}</ref> Simpler Improvements to overcome unwanted zero terms include an altered recurrence relation<ref>{{Cite journal|last1=Jaaskelainen|first1=T.|last2=Ruuskanen|first2=J.|date=1981-10-01|title=Note on Lentz's algorithm|url=http://dx.doi.org/10.1364/ao.20.003289|journal=Applied Optics|volume=20|issue=19|pages=3289–3290|doi=10.1364/ao.20.003289|pmid=20333144 |bibcode=1981ApOpt..20.3289J |issn=0003-6935}}</ref> suggested by Jaaskelainen and Ruuskanen in 1981 or a simple shift of the denominator by a very small number as suggested by Thompson and Barnett in 1986.<ref name=":1">{{Cite journal|last1=Thompson|first1=I.J.|last2=Barnett|first2=A.R.|date=1986|title=Coulomb and Bessel functions of complex arguments and order|url=http://dx.doi.org/10.1016/0021-9991(86)90046-x|journal=Journal of Computational Physics|volume=64|issue=2|pages=490–509|doi=10.1016/0021-9991(86)90046-x|bibcode=1986JCoPh..64..490T |issn=0021-9991}}</ref>
== Initial work ==
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