Modules

Compute nome $q$ for elliptic integrals corresponding to modulus $k$ $$\begin{aligned} K (k) &:= \int_{ 0 }^{ \pi / 2 } \frac{ d \theta }{ \sqrt{ 1 - { k }^{ 2 } \sin^{ 2 } \theta } } \\ { k }^{ \prime } &:= \sqrt{ 1 - { k }^{ 2 } } \\ { K }^{ \prime } (k) &:= K ( { k }^{ \prime } ) \\ q &:= e^{ - \pi { K }^{ \prime } (k) / K (k) } \\ \varepsilon &:= \frac{ 1 }{ 2 } \frac{ 1 - \sqrt{ { k }^{ \prime } } }{ 1 + \sqrt{ { k }^{ \prime } } } \end{aligned}$$

Compute nome $q$ for elliptic integrals corresponding to modulus $k$ $$\begin{aligned} K (k) &:= \int_{ 0 }^{ \pi / 2 } \frac{ d \theta }{ \sqrt{ 1 - { k }^{ 2 } \sin^{ 2 } \theta } } \\ { k }^{ \prime } &:= \sqrt{ 1 - { k }^{ 2 } } \\ { K }^{ \prime } (k) &:= K ( { k }^{ \prime } ) \\ q &:= e^{ - \pi { K }^{ \prime } (k) / K (k) } \\ \varepsilon &:= \frac{ 1 }{ 2 } \frac{ 1 - \sqrt{ { k }^{ \prime } } }{ 1 + \sqrt{ { k }^{ \prime } } } \end{aligned}$$