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→Lambda-star: Lambda-star in the complex plane doesn't appear to be studied in any reliable source |
→Modular equations: More general |
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Line 62:
:<math>u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0,</math>
:<math>(1-u^8)(1-v^8)-(1-uv)^8=0.</math>
The restriction to <math>\operatorname{Re}\tau=0</math> can be dropped if we [[Analytic continuation|analytically extend]] <math>u</math> and <math>v</math> by
:<math>u=\sqrt{2}e^{p\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2kp\pi i\tau}}{1+e^{(2k-1)p\pi i\tau}},\quad v=\sqrt{2}e^{\pi i\tau/8}\prod_{k=1}^\infty \frac{1+e^{2k\pi i\tau}}{1+e^{(2k-1)\pi i\tau}}.</math>
==Lambda-star==
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