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 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 J. C. P. MILLER' Con tents Mathematical Properties . . . . . . . . . . . . . . . . . . . . 686 Page 19.1. The Parabolic Cylinder Functions, Introductory . . . . . 686 d2Y- The Equation -2 (\$z2+a)y-0 dx 19.2 to 19.6. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations, Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . 686 19.7 to 19.11. Asymptotic Expansions . . . . . . . . . . . . . 689 19.12 to 19.15. Connections With Other Functions . . . . . . . 691 The Equation d2Y ~+(\$z"-a)Y=o 19.16 to 19.19. Power Series, Standard Solutions, Wronskian and Other Relations, Integral Representations . . . . . . . . 19.20 to 19.24. Asymptotic Expansions . . . . . . . . . . . . 693 19.25. Connections With Other Functions . . . . . . . . . . . 19.26. Zeros . . . . . . . . . . . . . . . . . . . . . . . . 696 19.27. Bessel Functions of Order hi, + 9 as Parabolic Cylinder Functions. . . . . . . . . . . . . . . . . . . . . 697 692 695 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 697 19.28. Use and Extension of the Tables . . . . . . . . . . . . 697 References . . . . . . . . . . . . . . . . . . . . . . . . . . 700 Table 19.1. U(a, z) and V ( a , z) ( 0 5 ~ 1 5 ) . . . . . . . . . . . 702 Table 19.2. W(a, &z) (05x55). . . . . . . . . . . . . . . . 712 Table 19.3. Auxiliary Functions . . . . . . . . . . . . . . . . . 720 -fa=0(.1)1(.5)5; ~=0(.1)5, 5s fa=0(.1)1(1)5; x=0(.1)5, 4-5D or S The author acknowledges permission from H.M. Stationery Office to draw freely from [19.11] the material in the introduction, and the tabular values of W(u, 2) for a=-5(1)5, fz=0(.1)5. Other tables of W(a, 5) and the tables of U(a, 5) and V ( a , 2) were prepared on EDSAC 2 a t the University Mathematical Laboratory, Cambridge, England, using a program prepared by Miss Joan Walsh for solution of general ,second order linear homogeneous differential equations with quadratic polynomial coeEcients. The auxiliary tables were prepared at the Computation Laboratory of the National Bureau of Standards. 1 The University Mathematic@ Laboratory, Cambridge, England. (Prepared under 685 contract with the National Bureau of Standards.) 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.