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→Relations to other elliptic functions: This article focuses on tau, the nome is unnecessary. Anyway, writing theta_2 in terms of tau and q at once would require specifying the branches for non-integer exponents. |
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:<math> \frac{1}{\big(\lambda(\tau)\big)^{1/4}}-\big(\lambda(\tau)\big)^{1/4} = \frac{1}{2}\left(\frac{\eta(\tfrac{\tau}{4})}{\eta(\tau)}\right)^4 = 2\,\frac{\theta_4^2(0,\tfrac{\tau}{2})}{\theta_2^2(0,\tfrac{\tau}{2})}</math>
where<ref name=C63>Chandrasekharan (1985) p.63</ref>
:<math>\theta_2(0,\tau) =
:<math>\theta_3(0,\tau) = \sum_{n=-\infty}^\infty
:<math>\theta_4(0,\tau) = \sum_{n=-\infty}^\infty (-1)^n
In terms of the half-periods of [[Weierstrass's elliptic functions]], let <math>[\omega_1,\omega_2]</math> be a [[fundamental pair of periods]] with <math>\tau=\frac{\omega_2}{\omega_1}</math>.
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