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:<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><ref group="note">This was done because the 8th root makes discontinuities in the complex plane.</ref> 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>
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==References==
===Notes===
{{reflist|group=note}}
===Other===
* {{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | ___location=New York | isbn=978-0-486-61272-0 | year=1972 | zbl=0543.33001 | url-access=registration | url=https://archive.org/details/handbookofmathe000abra }}
* {{citation | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | pages=108–121 }}
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