Q2.2.4Complex numbers: \(i^i\)
Determine the value of \(i^i\) as a real number, where \(i = \sqrt{-1}\).
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First let's get the "mathematical" answer: we can use Euler's formula, \(e^{ix} = \cos x + i\sin x\) with \(x=\pi/2\) to obtain \(e^{i\pi/2} = i\). Therefore,
\[i^i = e^{i^2\pi/2} = e^{-\pi/2} = 0.207879576\cdots.\]
This is the principal value: \(i^i\) is multi-valued and in general
\[i^i = e^{-\pi/2 + 2n\pi}, \;\; n=0,1,2,\cdots.\]
In the Python shell, the principal value is returned as a complex number object even though the answer is real:
>>> 1j**1j
(0.20787957635076193+0j)
but we can get the real part as a float by accessing its real attribute:
>>> (1j**1j).real
0.20787957635076193