# §29.5 Special Cases and Limiting Forms

 29.5.1 $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(0\right)=\mathop{b^{m}_{\nu}\/}% \nolimits\!\left(0\right)=m^{2},$
 29.5.2 $\mathop{\mathit{Ec}^{0}_{\nu}\/}\nolimits\!\left(z,0\right)=2^{-\frac{1}{2}},$
 29.5.3 $\displaystyle\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=\mathop{\cos\/}\nolimits\!\left(m(\tfrac{1}{2}\pi-z)\right),$ $m\geq 1$, $\displaystyle\mathop{\mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=\mathop{\sin\/}\nolimits\!\left(m(\tfrac{1}{2}\pi-z)\right),$ $m\geq 1$.

Let $\mu=\max{(\nu-m,0)}$. Then

 29.5.4 $\lim_{k\to 1-}\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)=\lim_{k\to 1% -}\mathop{b^{m+1}_{\nu}\/}\nolimits\!\left(k^{2}\right)=\nu(\nu+1)-\mu^{2},$
 29.5.5 ${\lim_{k\to 1-}\frac{\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}% \right)}{\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(0,k^{2}\right)}=\lim% _{k\to 1-}\frac{\mathop{\mathit{Es}^{m+1}_{\nu}\/}\nolimits\!\left(z,k^{2}% \right)}{\mathop{\mathit{Es}^{m+1}_{\nu}\/}\nolimits\!\left(0,k^{2}\right)}}=% \frac{1}{(\mathop{\cosh\/}\nolimits z)^{\mu}}\mathop{F\/}\nolimits\!\left({% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}% \atop\tfrac{1}{2}};{\mathop{\tanh\/}\nolimits^{2}}z\right),$ $m$ even,
 29.5.6 $\lim_{k\to 1-}\frac{\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}% \right)}{\left.\ifrac{d\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2% }\right)}{dz}\right|_{z=0}}=\lim_{k\to 1-}\frac{\mathop{\mathit{Es}^{m+1}_{\nu% }\/}\nolimits\!\left(z,k^{2}\right)}{\left.\ifrac{d\mathop{\mathit{Es}^{m+1}_{% \nu}\/}\nolimits\!\left(z,k^{2}\right)}{dz}\right|_{z=0}}=\frac{\mathop{\tanh% \/}\nolimits z}{(\mathop{\cosh\/}\nolimits z)^{\mu}}\mathop{F\/}\nolimits\!% \left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2},\tfrac{1}{2}\mu+\tfrac{1}{% 2}\nu+1\atop\tfrac{3}{2}};{\mathop{\tanh\/}\nolimits^{2}}z\right),$ $m$ odd,

where $\mathop{F\/}\nolimits$ is the hypergeometric function; see §15.2(i).

If $k\to 0+$ and $\nu\to\infty$ in such a way that $k^{2}\nu(\nu+1)=4\theta$ (a positive constant), then

 29.5.7 $\displaystyle\lim\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(\tfrac{1}{2}\pi-z,% \theta\right),$ $\displaystyle\lim\mathop{\mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$ $\displaystyle=\mathop{\mathrm{se}_{m}\/}\nolimits\!\left(\tfrac{1}{2}\pi-z,% \theta\right),$

where $\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(z,\theta\right)$ and $\mathop{\mathrm{se}_{m}\/}\nolimits\!\left(z,\theta\right)$ are Mathieu functions; see §28.2(vi).