1. [Taylor Series] Let us consider f(z) = exp(iz)/z² - 1, analytic on C\{-1, 1}.

(a) Write the first 5 terms of the Taylor expansion for f(z), centered at 0.

(b) Compute f⁽⁴⁾ (0).

(c) Without computing f'(z), write the first 4 terms of the Taylor expansion for f'(z). centered at 0. What is the radius of convergence for this series?

2. [Laurent Expansion] Let us consider f(z) = 1/z²(2 - 1).

(a) Decompose f(z) into partial fractions. That is, write f(z) as A/z + B/z² + C/z-1.

(b) Write the Laurent expansion of f(z) on the region 0 < |z| <1.

(c) Write the Laurent expansion of f(z) on the region 0 < lz - 1| < 1. Here, you may use the fact that if f (z) has Laurent expansion ∑^+∞_{n == ∞} a_nzⁿ, then f(z) has Lament expansion ∑^+∞_{n == ∞}n.a_nz^n-1.

3. [Isolated Singularity] Consider f(z) = sin(Π/2)/z⁴ +1

(a) Find all isolated singularities for f(z).

(b) Classify all isolated singularities for f(z). If it is removable, find a value at that singularity so that f(z) can be extended to an analytic function: if it is a pole. find the order of the pole.

4. [Essential Singularity] We have seen that the function behaves wildly around an essential singularity. Let us fix f(z) with an essential singularity at 2.0. We shall prove for any δ > 0, the range of f on the region 0 < |z - z₀l < δ is dense in C. Throughout this question, let us fix δ > 0. Suppose for the sake of a contradiction that f(z) cannot take value close to some b ∈ C.

(a) Suppose b ∈ C be such that there exists an  ∈ > 0 so that for all 0 < |z - z₀| < δ. |f(z) -b| > ∈. Define g(z) = 1/(f(z) - b) on 0 <| z -z₀|  < δ. Prove that g(z) is analytic on its domain. and is bmmded above by 1/∈.

(b) Deduce that z_o is a removable singularity for g(z).

(c) Prove that z_o is either a pole or a removable singularity for f(z). and thus there is a contradiction. Therefore. we reach the conclusion that the range of an analytic function near an essential singularity is dense.

(d) Consider f(z) = exp(1/z). Show that for any δ > 0,1(z) can take any value in C\(0) when z is in the domain 0 < |z| < δ. (Hint: show that |f(z)| can take any positive number first.)

(e) Prove that if z₀ is an essential singularity for f(z), then for any δ > 0, f(z) can take any value in C except possibly one exception, when 0 < |z - z₀| < δ. (Hint this is VERY hand)