F(z) = [1-cosh(z)] / z^3. its pole, order & residue

[bobmerhebi]

1. Show that f(z) = [1 - cosh(z)] / z³ has a pole as its singular point. Determine its order m & find the residue B.


2.

lim |f(z)| tends to $$\infty$$ as z tends to singular point

b_m + b_m+1(z-z₀)+...+ b₁(z-z₀)^m-1 + $$\sum$$$$^{\infty}_{n = 0}$$a_n(z-z₀)ⁿ= (z-z₀) ^m f(z)


3. f(z) = [1 - cosh(z)] / z³ = 1/z³ - cos(iz)/z³

that's all i could do. I don't know how to do the limit of the cos part. does it approach infinity ? I think NOt right ?

please help asap.

thx in advnace

 

[eok20]

Expand out cosh(z) = cos(iz) using its Taylor series.

 

[bobmerhebi]

Expand out cosh(z) = cos(iz) using its Taylor series.

Doing this i get:

f(z) = 1 - (1 + z²/2 + z⁴ / 4! + z⁶ / 6! + ...)

= - $$\sum^{\infty}_{n = 1}$$z²ⁿ/(2n)! tends to 0 as z tends to 0

but this is not the limit we have for a pole. but its for a removable singular point. isn't it ?