1.5. The Euler formula

Eulers identity

(1.24)\[e^{i\pi}+1=0\]

Eulers formula

(1.25)\[e^{ix}=\cos x+i\sin x\]

**Cosine exponentialform

(1.26)\[\frac{e^{ix}+e^{-ix}}{2}=\cos x\]

Sine exponential form

(1.27)\[\frac{e^{ix}-e^{-ix}}{2i}=\sin x\]

Tangent exponential form

(1.28)\[\frac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-1x})}=\tan x\]

Complex exponential

(1.29)\[e^{x+iy}=e^x(\cos y+i\sin y)\]

de Moivre’s theorem

(1.30)\[(\cos x+i\sin x)^n=\cos nx+i\sin nx\]

You can read more about this in Adams and Essex [AE18].