- #1

- 10

- 0

**Question 1:**

Find the Laurent series of [itex]\cos{\frac{1}{z}}[/itex] at the singularity [itex]z = 0[/itex].

The answer is often given as,

[tex]\cos\frac{1}{z} = 1 - \frac{1}{2z^2} + \frac{1}{24z^4} - ...[/tex]

Which is the MacLaurin series for [itex]\cos{u}[/itex] with [itex]u = \frac{1}{z}[/itex]. The MacLaurin series is the Taylor series when [itex]u_0 = 0[/itex], however, we are interested in the "point" where [itex]u_0 = \infty[/itex]! Why is this answer considered valid if it expands the function around the wrong point?

**Question 2:**

If the above answer is considered correct then if I was interested in finding the Laurent series of [itex]\cos{\frac{1}{z-1}}[/itex] at the singularity of [itex]z=1[/itex] then would the answer simply be the Taylor expansion of [itex]\cos{u}[/itex] around the point [itex]u_0 = [/itex]1 with [itex]u = \frac{1}{z}[/itex]?