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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: 20. Mathieu Functions Mathematical Properties 20.1. Mathieu’s Equation Canonical Form of the Differential Equation 20.1.1 Mathieu’s Modified Differential Equation 20.1.2 -2 d?f - (a-2q cosh 2u)f=O (v=iu, y=f) d u Relation Between Mathieu’s Equation and the Wave Equation for the Elliptic Cylinder The wave equation in Cartesian coordinates is b2W b2W b2W+k2W,0 20.1.3 -+2+= 3x2 b y A solution W is obtainable by separation of vari- ables in elliptical coordinates. Thus, let x=p cosh u cos v; y=p sinh u sin v; z=z; p a positive constant; 20.1.3 becomes 20.1.4 * b2W’ 2 2w b2W dz2 p2 (cosh 2u-cos 2v) (G+m)+k2w=o Assuming a solution of the form w= c.(z)f(u>s(v) and substituting the above into 20.1.4 one obtains, after dividing through by W, where 2 * G=- p2 (cosh 2u-cos 2;) Since 2, u, v are independent variables, it follows that d2c. - 20.1.5 dz2+CV’ where c is a constant. Again, from the fact that G=c and that u, v are independent variables, one sets 20.1.6 * where a is a constant. The above are equivalent to 20.1.1 and 20.1.2. The constants c and a are often referred to as separation constants, due to the role they play in 20.1.5 and 20.1.6. For some physically important solutions, the function g must be periodic, of period ?r or 2 ~ . It can be shown that there exists a countably infinite set of characteristic values a,(q) which yield even periodic solutions of 20.1.1; there is another countably infinite sequence of characteristic values b,(q) which yield odd periodic solutions of 20.1.1. It is known that there exist periodic solutions of period k?r, where k is any positive integer. In what follows, however, the term characteristic value will be reserved for a value associated with solutions of period ?r or 2?r only. These character- istic values are of basic importance to the general theory of the differential equation for arbitrary parameters a and q. An Algebraic Form of Mathieu’s Equation 20.1.7 (1 - t2) 9 - t dY -+ (a+ 2q-4qt2) y=O dt2 dt (cos v=t) Relation to Spheroidal Wave Equation 20.1.8 * Thust Mathieu’s equation is a special case of 20.1.8, with E = - # , c=a+2q. 20.2. Determination of Characteristic Values A solution of 20.1.1 with v replaced by z, having period ?r or 2?r is of the form m 20.2.1 y = c m=O (Am cos mz+ B, sin mz) where B, can be taken as zero. substituted into 20.1.1 one obtains 20.2.2 If the above is *See page n. 722 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.