Complex Analysis: Taylor's Theorem

[tylerc1991]

Homework Statement

Find the Maclaurin series representation of:

f(z) = {sinh(z)/z for z =/= 0 }
{0 for z = 0 }

Note: wherever it says 'sum', I am noting the sum from n=0 to infinity.

The Attempt at a Solution

sinh(z) = sum [z^(2n+1)/(2n+1)!]

=> sinh(z)/z = sum [z^(2n)/(2n+1)!] (referenced as (1))

=> the Maclaurin series representation for f(z) is (1) when z =/= 0 and 0 when z=0

Thank you for the help, I hope the text is not confusing.

 

[micromass]

Correct, but

the Maclaurin series representation for f(z) is (1) when z =/= 0 and 0 when z=0

The distinction is not necessary in this case. If z=0, then (1) already yields 0. Thus (1) is the series representation on the entire domain!