The Ince equation is the linear second-order ordinary differential equation
![w^('')(z)+xisin(2z)w^'(z)+[eta-pxicos(2z)]w(z)=0,](/images/equations/InceEquation/NumberedEquation1.svg) |
(1)
|
where primes denote derivatives with respect to 
and 
, 
, and 
are constants (NIST DLMF). The substitution ![W(z)=w(z)exp[-xicos(2z)/4]](/images/equations/InceEquation/Inline5.svg)
gives the Whittaker-Hill
differential equation
![W^('')(z)+[eta-(xi^2)/8-(p+1)xicos(2z)+(xi^2)/8cos(4z)]W(z)=0.](/images/equations/InceEquation/NumberedEquation2.svg) |
(2)
|
When 
is a nonnegative integer and 
is real, suitable values of 
give solutions that are trigonometric
polynomials. These solutions, called Ince polynomials, are finite Fourier
series with harmonics at most 
and of the same parity as 
. For example, when 
,
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(3)
|
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(4)
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