New Online Book! Handbook of Mathematical Functions (AMS55)
Conversion & Calculation Home >> Reference Information
Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
Purchase the electronic edition of this book in Adobe PDF format!
FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE
| ABOUT | SMALL | MEDIUM | LARGE
Below is the OCR-scanned text from this page:
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
The page scan image above, and the text in the text box above,
are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.