Partner with ConvertIt.com
 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: PARABOLIC CYLINDER FUNCTIONS 687 19.3. Standard Solutions These have been chosen to have the asymptotic behavior exhibited in 19.8. The first is Whit- taker's function [19.8, 19.91 in a more symmetrical no tation. 19.3.1 U(a, 2) =D+*(Z) =cos?r(~+~)*Y1 -sin7r(+++a) .Y2 19.3.2 in which 19.3.6 2*+1 sin?r(%-#a) r (t - +a) V(a, 0) = In terms of the more familiar D,(s) of Whit- taker, 19.3.1 U(a, 2) =D-,-*(z) 19.3.8 1 V ( U , 2) =- r (++a) {sin TU.D-,-~(Z) +D-,-t(-z> 1 7r 19.4. Wronskian and Other Relations 19.4.1 W{U,V) =a 19.4.2 TV(U, Z) = r (++a) {sin m.U(a, 2) +U(U, -4 } 19.4.3 r(++a)u(a, z)=r sec2 ra(V(a, -4 -sin ra.V(u, z) 1 19.4.4 r(t-+a) cos .-(\$++a> fiz*-: y1 = 2 sin n(B+ +a). Y =U(a, x) +U(a, -2) 19.4.5 r (9 -+a) sin .-(\$+\$a) - @,a-t yz=2 cos 7r(\$++a).Y, =U(a, 2) -U(a, -2) 19.4.6 &GU(--a, fix)= r(++a) (e-*r(*-f)U(a, f~)+e**(+~-t)U(a, FZ)} 19.4.7 &U(a, &2)= r(+-a) { e-**(*+W(-a, fiz)+e*"(*+t'U(--a, Fix)} 19.5. Integral Representations A full treatment is given in [19.11] section 4. Representations are given here for U(a, z ) only; othersmay bederived by useof therelationsgiven in 19.4. 19.5.2 where (Y and 19.1 and 19.2. become indeterminate; in this case are the contours shown in Figures When a++ is a positive integer these integrals 1 19.5.3 U(a, z)=? e-fza ( s + 4 S PLANE a t PLANE FIGURE 19.1 --rr 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.