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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)!

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