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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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20. Mathieu Functions
20.1. Mathieu’s Equation
Canonical Form of the Differential Equation
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
* b2W’ 2 2w b2W
dz2 p2 (cosh 2u-cos 2v) (G+m)+k2w=o
Assuming a solution of the form
and substituting the above into 20.1.4 one obtains,
after dividing through by W,
2 * G=-
p2 (cosh 2u-cos 2;)
Since 2, u, v are independent variables, it follows
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
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
(1 - t2) 9 - t dY -+ (a+ 2q-4qt2) y=O
dt2 dt (cos v=t)
Relation to Spheroidal Wave Equation
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
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
If the above is
*See page n. 722
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