The Mathieu functions
MathieuC
and
MathieuS
are solutions to the equation
. This equation appears in many physical situations that involve elliptical shapes or periodic potentials. The function
MathieuC is defined to be even in
, while
MathieuS is odd.
When
the Mathieu functions are simply
and
. For nonzero
, the Mathieu functions are only periodic in
for certain values of
. Such Mathieu characteristic values are given by
MathieuCharacteristicA
and
MathieuCharacteristicB
with
an integer or rational number. These values are often denoted by
and
.
For integer
, the even and odd Mathieu functions with characteristic values
and
are often denoted
and
, respectively. Note the reversed order of the arguments
and
.
According to Floquet's theorem, any Mathieu function can be written in the form
, where
has period
and
is the Mathieu characteristic exponent
MathieuCharacteristicExponent
. When the characteristic exponent
is an integer or rational number, the Mathieu function is therefore periodic. In general, however, when
is not a real integer,
and
turn out to be equal.
for even Mathieu functions with characteristic exponent r and parameter q
for odd Mathieu functions with characteristic exponent r and parameter q
