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Let $Z=-4i$ That is $Z=4[0+(-1)i]$ You have to find a theta value such that $\sin \theta =-1$ and $\cos \theta =0 $ Since $\cos \frac{3\pi}{2}=0$ and $\sin \frac{3\pi}{2}=-1 $ Thus $$Z=4 \left[\cos \frac{3\pi}{2}+i \sin \frac{3\pi}{2} \right]$$ Since $$e^{i\theta}=\cos \theta +i \sin \theta$$ $$Z=4 e^{i\frac{3\pi}{2}}$$


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Have you tried any examples? Simplest example to try: $P(z)=z$.


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The Laurent series at $0$ is defined with the help of the Bernoulli numbers.



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