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 Mathematical Properties 19.1. The Parabolic Cylinder Functions Introductory 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 dx2 The functions 19.1.4 ?/(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: 19.1.5 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 686 19.2.1 = e-txz1 Fl (+a++; 6; +x2) 19.2.2 =e'z2M(-+u+\$, 3, +x2) 19.2.4 =xetx2M(-+u+s, Q, -+x2) these series being conmrgent for all values of x (see chapter 13 for M(a, c, 2)). Alternatively , 19.2.5 19.2.6 y2=x+u -+ 53 ( u2+- ;)\$+(u3+yu)\$ 3! +( U4+17d+~) \$f(a6+35a3+~ a) \$+ in which non-zero coefficients (I,, of x"/n! are connected by 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. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.