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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 | MEDIUM | LARGE Below is the OCR-scanned text from this page: MATHIEU FUNCTIONS 723 Equation 20.2.2 can be reduced to one of four simpler types, given in 20.2.3 and 20.2.4 below m 20.2.3 yo=C Azm+* cos (2m+p)z, p=O or 1 m=O m 20.2.4 y l = C sin (2m+p)z, p=O or 1 If p=O, the solution is of period a; if p=l, the m=O solution is of period 2a. Recurrence Relations Among the Coefficients Even solutions of period a: 20.2.5 uAO- qA2 = 0 20.2.6 20.2.7 (a - 4) A2 - Q (2Ao + Ah) = 0 (a- mZ) Am- n(Am-z+Arn+z) = O Even solutions of period 2a: 20.2.8 (~-1)Ai-p(Ai+Ad = O . along with 20.2.7 for m 2 3 . Odd solutions of period a: 20.2.9 ( U - 4 ) B z - @4= 0 * 20.2.10 (U--~)B,-~((B,-,+B,+~)=O Odd solutions of period 2a: 20.2.11 (a-1)Bi+q(Bi-B3)=Ol along wit,h 20.2.10 for m23. Let 20.2.12 Ge,=A,/A,-2, Gom= Bm/Bm-.z; G,=Gem or Go, when the same operations apply to both, and no ambiguity is likely to arise. Further let 20.2.13 V,= (a-m2)/q. Equations 20.2.5-20.2.7 are equivalent to 2 20.2.14 Gez= Vo; Ge4= V2-- 20.2.15 G,= I/( V,- Gm+2) (m2 3) , for even solutions of period a. 20.2.16 Za, along with 20.2.15 20.2.17 2a, along with 20.2.15 Ge2 Similarly V,-l=Ge,; for even solutions of period Vl+l=Go,, for odd solutions of period *See page n. 20.2.18 along with 20.2.15 V2=Go4, for odd solutions of period T, These three-term recurrence relations among the coefficients indicate that every G, can be developed into two types of continued fractions. Thus 20.2.15 is equivalent to 20.2.19 1 . . . (m23) 1 1 =--- 1 G,= Vm-Gm+z Vm- Vrni2- Vmid- 20.2.20 Grn+z=Vm- 1/Gm where pl =d= 0 ; cpo= 2, if Gm+z= Azs/Azs-2 (p,=d= cpo=O, if Gm+z=B2s/B2s-z n=d=cpo=l, if Gm+z=Bza+l/Bza-l n=-l; Ipo=d=1, if Gm+z=A28+1/A2s-i The four choices of the parameters (q, ‘pol d correspond to the four types of solutions 20.2.3- 20.2.4. Hereafter, it will be convenient to sep- arate the characteristic values a into two major subsets: a=a,, associated with even periodic solutions a = b,, associated with odd periodic solutions If 20.2.19 is suitably combined with 20.2.13-20.2.18 there result four types of continued fractions, the roots of which yield the required characteristic values 20.2.22 20.2.24 . . . =o V,+l-- - - 1 1 1 v3- v6- v,- If a is a root of 20.2.21-20.2.24, then the Corre- sponding solution exists and is an entire function of z, for general complex values of p. If p is real, then the Sturmian theory of second order linear differential equations yields the 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.