📂Complex Anaylsis

The Relationship Between the Powers of i and the Powers of e

Theorem

Natural constant $e$ and imaginary number $i$ raised to a power satisfy the following relationship.

$$e^{i\frac{l \pi}{2}} = i^{l}$$

Proof

Since $e^{ i \frac{l \pi}{2}}=\cos\frac{l \pi}{2}+i\sin \frac{l \pi}{2}$, when $l=0$,

$$e^{ 0}= 1 =i^{0}$$

When $l=1$,

$$e^{ i \frac{\pi}{2}}=\cos\frac{\pi}{2}+i\sin \frac{\pi}{2}=i=i^{1}$$

When $l=2$,

$$e^{ i \pi}=\cos \pi+i\sin \pi=-1=i^{2}$$

When $l=3$,

$$e^{ i \frac{3\pi}{2}}=\cos\frac{3\pi}{2}+i\sin \frac{3\pi}{2}=-i=i^{3}$$

As it repeats thereafter,

$$e^{ i \frac{l \pi}{2}}=i^l$$

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