The differential equation for θ translates into differential equations for both the functions mentiooned on the previous page:
4(1 + T2)(T' + zT") = 1 − 6T2 + T4 + 8TT'2 and zww" + ww' − zw'2 = i(1 + w4)/4.
Although both equations are more complicated looking than the original, they are both algebraic (i.e. they do not include any transcendental functions like the cosine). The two functions are more closely related than one might expect:
T = −i(w − 1)/(w + 1) and w = (1 + iT)/(1 − iT).
These are linear fractional transformations, which are generalized rotations of the Riemann sphere (the complex plane with a point at infinity), so we can use whichever is more convenient to get information about θ. T has the advantage that it is an odd function whose Taylor coefficients are real, so it takes real values on the real axis and imaginary values on the imaginary axis. But graphing it runs into problems when 1 + T2 = 0, since this blocks solving for T". This difficulty occurs at precisely the singularities of θ. Although it has poles, they do not cause a problem because we can pass to the cotangent, which satisfies a similar equation.
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