In mathematics, the Ince equation, named for Edward Lindsay Ince, is the differential equation
w
′
′
+
ξ
sin
(
2
z
)
w
′
+
(
η
−
p
ξ
cos
(
2
z
)
)
w
=
0.
{\displaystyle w^{\prime \prime }+\xi \sin(2z)w^{\prime }+(\eta -p\xi \cos(2z))w=0.\,}
When p is a non-negative integer, it has polynomial solutions called Ince polynomials. In particular, when
p
=
1
,
η
±
ξ
=
1
{\displaystyle p=1,\eta \pm \xi =1}
, then it has a closed-form solution
w
(
z
)
=
C
e
−
i
z
(
e
2
i
z
∓
1
)
{\displaystyle w(z)=Ce^{-iz}(e^{2iz}\mp 1)}
where
C
{\displaystyle C}
is a constant.
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