Modular Equations and Approximations to π

摘要: If we suppose that $$(1 + {e^{ - \pi Nn}})(1 3\pi b\pi Nn}}) \ldots = {2^{\frac{1}{4}}}{c^{ Nn/24}}{G_n} $$ (1) and {2^{\frac{1}{4}}}{e^{ Nn/24}}{g_n}, $$ (2) then G n and g can always be expressed as roots of algebraical equations when is any rational number. For know q)(1 {q^3})(1 {q^5}) {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{(kk')^{ \frac{1}{{12}}}} $$ (3) and {2^{\frac{1}{6}}}{q^{\frac{1}{{24}}}}{k^{ \frac{1}{{12}}}}{k'^{\frac{1}{6}}}. $$ (4)