In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper (Gabriel Lamé 1837). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.
The Lamé equation Lamé's equation is
d 2 y d x 2 + ( A + B ℘ ( x ) ) y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}+(A+B\wp (x))y=0,}
where A and B are constants, and ℘ {\displaystyle \wp } is the Weierstrass elliptic function. The most important case is when B ℘ ( x ) = − κ 2 sn 2 x {\displaystyle B\wp (x)=-\kappa ^{2}\operatorname {sn} ^{2}x} , where sn {\displaystyle \operatorname {sn} } is the elliptic sine function, and κ 2 = n ( n + 1 ) k 2 {\displaystyle \kappa ^{2}=n(n+1)k^{2}} for an integer n and k {\displaystyle k} the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points. By changing the independent variable to t {\displaystyle t} with t = sn x {\displaystyle t=\operatorname {sn} x} , Lamé's equation can also be rewritten in algebraic form as
d 2 y d t 2 + 1 2 ( 1 t − e 1 + 1 t − e 2 + 1 t − e 3 ) d y d t − A + B t 4 ( t − e 1 ) ( t − e 2 ) ( t − e 3 ) y = 0 , {\displaystyle {\frac {d^{2}y}{dt^{2}}}+{\frac {1}{2}}\left({\frac {1}{t-e_{1}}}+{\frac {1}{t-e_{2}}}+{\frac {1}{t-e_{3}}}\right){\frac {dy}{dt}}-{\frac {A+Bt}{4(t-e_{1})(t-e_{2})(t-e_{3})}}y=0,}
which after a change of variable becomes a special case of Heun's equation. A more general form of Lamé's equation is the ellipsoidal equation or ellipsoidal wave equation which can be written (observe we now write Λ {\displaystyle \Lambda } , not A {\displaystyle A} as above)
d 2 y d x 2 + ( Λ − κ 2 sn 2 x − Ω 2 k 4 sn 4 x ) y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}+(\Lambda -\kappa ^{2}\operatorname {sn} ^{2}x-\Omega ^{2}k^{4}\operatorname {sn} ^{4}x)y=0,}
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