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29 Lamé FunctionsLamé Functions

§29.5 Special Cases and Limiting Forms

29.5.1 aνm(0)=bνm(0)=m2,
29.5.2 𝐸𝑐ν0(z,0)=212,
29.5.3 𝐸𝑐νm(z,0) =cos(m(12πz)),
𝐸𝑠νm(z,0) =sin(m(12πz)),

Let μ=max(νm,0). Then

29.5.4 limk1aνm(k2)=limk1bνm+1(k2)=ν(ν+1)μ2,
29.5.5 limk1𝐸𝑐νm(z,k2)𝐸𝑐νm(0,k2)=limk1𝐸𝑠νm+1(z,k2)𝐸𝑠νm+1(0,k2)=1(coshz)μF(12μ12ν,12μ+12ν+1212;tanh2z),
m even,
29.5.6 limk1𝐸𝑐νm(z,k2)d𝐸𝑐νm(z,k2)/dz|z=0=limk1𝐸𝑠νm+1(z,k2)d𝐸𝑠νm+1(z,k2)/dz|z=0=tanhz(coshz)μF(12μ12ν+12,12μ+12ν+132;tanh2z),
m odd,

where F is the hypergeometric function; see §15.2(i).

If k0+ and ν in such a way that k2ν(ν+1)=4θ (a positive constant), then

29.5.7 lim𝐸𝑐νm(z,k2) =cem(12πz,θ),
lim𝐸𝑠νm(z,k2) =sem(12πz,θ),

where cem(z,θ) and sem(z,θ) are Mathieu functions; see §28.2(vi).