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Reference

$\left({\frac {g}{h}}\right)'={\frac {g'h-gh'}{h^{2}}}$

$\left({\frac {g}{h}}\right)''={\frac {(g''h-gh'')h-2(g'h-gh')h'}{h^{3}}}={\frac {g''h^{2}-(2g'h'+gh'')h+2gh'^{2}}{h^{3}}}$

$e^{ab}=\left(e^{a}\right)^{b}$

$e^{ix+y}=e^{x}(\cos {y}+i\sin {y})$

$\operatorname {sn} (u+v)={\frac {\operatorname {sn} u\operatorname {cn} v\operatorname {dn} v+\operatorname {cn} u\operatorname {dn} u\operatorname {sn} v}{1-k^{2}\operatorname {sn} ^{2}u\operatorname {sn} ^{2}v}}$

$\operatorname {cn} (u+v)={\frac {\operatorname {cn} u\operatorname {cn} v-\operatorname {sn} u\operatorname {dn} u\operatorname {sn} v\operatorname {dn} v}{1-k^{2}\operatorname {sn} ^{2}u\operatorname {sn} ^{2}v}}$

$\operatorname {dn} (u+v)={\frac {\operatorname {dn} u\operatorname {dn} v-k^{2}\operatorname {sn} u\operatorname {cn} u\operatorname {sn} v\operatorname {cn} v}{1-k^{2}\operatorname {sn} ^{2}u\operatorname {sn} ^{2}v}}$

Pe

$\wp (z)=z^{-2}+{\frac {1}{20}}g_{2}z^{2}+{\frac {1}{28}}g_{3}z^{4}+{\mathcal {O}}(z^{6})$

$\wp '(z)=-2z^{-3}+{\frac {1}{10}}g_{2}z+{\frac {1}{7}}g_{3}z^{3}+{\mathcal {O}}(z^{5})$

$\wp ''(z)=6z^{-4}+{\frac {1}{10}}g_{2}+{\frac {3}{7}}g_{3}z^{2}+{\mathcal {O}}(z^{4})$

$\wp ^{2}(z)=z^{-4}+{\frac {2}{20}}g_{2}+{\frac {2}{28}}g_{3}z^{2}+{\mathcal {O}}(z^{4})$

$\wp ''(z)-6\wp ^{2}(z)=-{\frac {1}{2}}g_{2}+{\mathcal {O}}(z^{4})$

${y=\wp (z)}\Leftrightarrow {y''=4y^{2}-{\frac {1}{2}}g_{2}y}$