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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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19. Parabolic Cylinder Functions
19.1. The Parabolic Cylinder Functions
These are solutions of the differential equation
19.1.1 dx2 Q!+(ax2+bx+c)y=O
19.1.3 $$+ (~x"-a)y=O
with two real and distinct standard forms
19.1.2 d2?J- (tx"+a)y=O
?/(a, 2) !/(a, -5) Y(-% ix) Y(-Q, -k)
are all solutions either of 19.1.2 or of 19.1.3 if any
one is such a solution.
Replacement of a by --ia and x by xdf* converts
19.1.2 into 19.1.3. If y(a, x) is a solution of 19.1.2,
then 19.1.3 has solutions:
y(-ia, xet'") y(-ia, -mif")
y(ia, -xe-**") y(ia, xe-tfr)
Both variable x and the parame-x a may take
on general complex values in this section and in
many subsequent sections. Practical applications
appear to be confined to real solutions of real equa-
tions; therefore attention is confined to such solu-
tions, and, in general, formulas are given for the
two equations 19.1.2 and 19.1.3 independently.
The principal computational consequence of the
remarks above is that reflection in the y-axis
produces an independent solution in almost all
cases (Hermite functions provide an exception),
so that tables may be confined either to positive
x or to a single solution of 19.1.2 or 19.1.3.
The Equation *-(: dx2 e+,> y=o
19.2. Power Series in x
Even and odd solutions of 19.1.2 are given by
= e-txz1 Fl (+a++; 6; +x2)
=e'z2M(-+u+$, 3, +x2)
=xetx2M(-+u+s, Q, -+x2)
these series being conmrgent for all values of x
(see chapter 13 for M(a, c, 2)).
y2=x+u -+ 53 ( u2+- ;)$+(u3+yu)$ 3!
+( U4+17d+~) $f(a6+35a3+~ a) $+
in which non-zero coefficients (I,, of x"/n! are
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