Other pairs of independent solutions may be formed from linear combinations of the above solutions.[2] One such pair is based upon their behavior at infinity:
where
The function U(a, z) approaches zero for large values of z and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z .
and
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902))[3] that are themselves sometimes called parabolic cylinder functions:[2]
Function Da(z) was introduced by Whittaker and Watson as a solution of eq.~(1) with bounded at .[4] It can be expressed in terms of confluent hypergeometric functions as
Power series for this function have been obtained by Abadir (1993).[5]
^Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" Proc. London Math. Soc., 35, 417–427.
^Whittaker, E. T. and Watson, G. N. (1990) "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348.
^ Abadir, K. M. (1993) "Expansions for some confluent hypergeometric functions." Journal of Physics A, 26, 4059-4066.
^NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.2 of 2024-09-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
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