Apuntes de Variable Compleja - Carlos Lizama homepage ...
Apuntes de Variable Compleja - Carlos Lizama homepage ...
Apuntes de Variable Compleja - Carlos Lizama homepage ...
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3.2. REPRESENTACIONES POR SERIES DE TAYLOR 19<br />
Entonces<br />
3. Función coseno:<br />
cos z =<br />
∞<br />
n=0<br />
sin z =<br />
∞<br />
n=0<br />
(−1) n z 2n+1<br />
(2n + 1)!<br />
(−1) n (2n + 1)z 2n<br />
(2n + 1)(2n)! =<br />
∞<br />
n=0<br />
(−1) n z 2n<br />
(2n)!<br />
4. Función seno hiperbólico:<br />
Sabemos que<br />
sinh iz = i sin z, para todo z, en particular para w = iz, entonces sinh w =<br />
i sin −iw<br />
Por lo tanto<br />
Analicemos<br />
Entonces<br />
sinh z = i sin −iz = i<br />
∞<br />
n=0<br />
(−1) n (−iz) 2n+1<br />
(2n + 1)!<br />
(−iz) 2n+1 = (−i) 2n+1 z 2n+1 = (−1) 2n (−1)[i 2 ] n iz 2n+1<br />
= (−1)(−1) n iz 2n+1 = (−i)(−1) n z 2n+1<br />
sinh z = −i 2<br />
5. Función coseno hiperbólico:<br />
cosh z =<br />
∞<br />
n=0<br />
=<br />
∞<br />
n=0<br />
∞<br />
n=0<br />
(−1) n (−1) n z 2n+1<br />
z 2n+1<br />
(2n + 1)!<br />
(2n + 1)!<br />
(2n + 1)z 2n<br />
(2n + 1)(2n)! =<br />
∞<br />
n=0<br />
z 2n<br />
(2n)!