: Jacobian elliptic function,
: derivative of 
with respect to 
,
: complex variable,
: real parameter,
: real parameter,
: real parameter and
: solution
where and are real parameters such that 
and
. For see §22.2.
This equation has regular singularities at the points
, where 
, and
, 
are the complete elliptic integrals of the
first kind with moduli , 
, respectively; see
§19.2(ii). In general, at each singularity each solution of
(29.2.1) has a branch point (§2.7(i)). See Figure
29.2.1.
![\begin{picture}(10.0,8.0)(-5.0,-4.0)\put(0.0,-4.0){\line(0,1){8.0}}
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\times$}\put(1.83,-3.15){$\times$}\put(3.83,-3.15){$\times$}
\put(-4.17,-1.15){$\times$}\put(-2.17,-1.15){$\times$}\put(-0.17,-1.15){$%
\times$}\put(1.83,-1.15){$\times$}\put(3.83,-1.15){$\times$}
\put(-4.17,0.85){$\times$}\put(-2.17,0.85){$\times$}\put(-0.17,0.85){$\times$}%
\put(1.83,0.85){$\times$}\put(3.83,0.85){$\times$}
\put(-4.17,2.85){$\times$}\put(-2.17,2.85){$\times$}\put(-0.17,2.85){$\times$}%
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{\put(-4.0,0.0){\line(0,1){0.15}}\put(-2.0,0.0){\line(0,1){0.15}}\put(0.0,0.0)%
{\line(0,1){0.15}}\put(2.0,0.0){\line(0,1){0.15}}\put(4.0,0.0){\line(0,1){0.15%
}}}
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}}}\/}\nolimits}\!$}
\put(-1.3,-1.15){$-\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{{\prime}%
}}\/}\nolimits}\!$}
\put(-0.9,0.85){$\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{{\prime}}}%
\/}\nolimits}\!$}
\put(-1.1,1.85){$2\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{{\prime}}%
}\/}\nolimits}\!$}
\put(-1.1,2.85){$3\iunit\!\DUAL[ATOM]{\CompEllIntCK@{k}}{\mathop{{K^{{\prime}}%
}\/}\nolimits}\!$}
\put(-4.6,-0.5){$-4\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(-2.6,-0.5){$-2\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(1.8,-0.5){$2\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\put(3.8,-0.5){$4\!\DUAL[ATOM]{\CompEllIntK@{k}}{\mathop{K\/}\nolimits}\!$}
\end{picture}](./29/2/F1.png)
-plane: singularities 
of Lamé’s equation.
: Legendre’s complementary complete elliptic integral of the first kind,
: Legendre’s complete elliptic integral of the first kind,
: complex variable and
: real parameter
: derivative of with respect to ,
: real parameter,
: real parameter,
: real parameter and
: variable
where
: Jacobian elliptic function,
: complex variable,
: real parameter and
: variable
: cosine function,
: derivative of with respect to ,
: sine function,
: real parameter,
: real parameter,
: real parameter and
: change of variable
where
: Jacobi’s amplitude function,
: complex variable,
: real parameter and
: change of variable
For 
see §22.16(i).
Next, let 
be any real constants that satisfy 
and
: real parameter and
: constants
(These constants are not unique.) Then with
: real parameter,
: real parameter,
: constants and
: Legendre’s complementary complete elliptic integral of the first kind,
: complex variable,
: real parameter,
: constants and
we have
(= 
= 
): Weierstrass 
-function,
: derivative of with respect to ,
: real parameter,
and
and
: derivative of with respect to ,
: real parameter,
: constants,
and
where
: Weierstrass -function,
(= = ): Weierstrass -function,
,
and
with
: constants and
For the Weierstrass function see §23.2(ii).
