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4.33 Complex trigonometric series

Euler’s formulas

(i) $e^{it}=\cos t+i\sin t$ ;

(ii) $e^{i\pi}+1=0$ ;

(iii) $e^{i\theta}=1$ if and only if $\theta=2\pi n$ for some integer $n.$

(i) From Maclaurin’s series we have

\cos t=1-{{t^{2}}\over{2!}}+{{t^{4}}\over{4!}}-\dots

\sin t=t-{{t^{3}}\over{3!}}+{{t^{5}}\over{5!}}-\dots

so we add these to get

\cos t+i\sin t=1+it-{{t^{2}}\over{2!}}-i{{t^{3}}\over{3!}}+\dots=\sum_{n=0}^{% \infty}{{(it)^{n}}\over{n!}}=e^{it}.