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The former expression, q^(1/24) led to some problems: 1. Take tau=1/2 + i * sqrt(3)/2. Then -1/tau = -1/2 + i * sqrt(3/2) = tau - 1. Note that q corresponding to that tau is -exp(-pi*sqrt(3)), while the q corresponding to -1/tau = tau - 1 is the exact same value. Thus the
Dedekind eta function has
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I think the definition of eta shown here was wrong and instead of q^(1/24) it should be exp(i pi tau / 12) as I've made it. If the latter expression was implied in the first place, then that should be mentioned since the two expressions are ordinarily different.
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the same value at both tau and -1/tau. But then the functional equation says that eta(-1/tau) = sqrt(-i*tau) * eta(tau), i.e. 1 = sqrt(-i*tau). This is incorrect. 1 is not the square root of (-i/2 + sqrt(3)/2).
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If You have ever seen the function as a correct 3D complex surface plot, You would not ask. So I want to impress and show a plot by
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The text mentions that eta is a modular function of weight 1/2. It'd be useful to include its level, and possibly character as well.
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2. Another problem is when tau = -1/2 + 0.01i, then eta(tau+1) = eta(tau) and the exp(i pi/12) factor couldn't be correct.
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The general modular transformation gives the eta function an argument z on the r.h.s., \tau would be more readable.
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Yes, but that's the standard notation. Besides, ad-bc=1 so the four args aren't independent anyway.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Strictly speaking, if the
Dedekind eta function is seen as the 24th root of
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Dedekind eta function over the complex numbers (high resolution)
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instead of q^{1/24}, and I have made the necessary changes.
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582:{\displaystyle \tau ={\tfrac {-1+{\sqrt {-3}}}{2}}}
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529:{\displaystyle \tau ={\tfrac {1+{\sqrt {-3}}}{2}}}
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444:{\displaystyle {\tfrac {\Delta }{(2\pi )^{12}}}}
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479:{\displaystyle \eta ({\tau })}
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