代写MAST30021 Complex Analysis, Semester 1 2025 Written assignment 3帮做R程序

School of Mathematics and Statistics

MAST30021 Complex Analysis, Semester 1 2025

Written assignment 3 and Cover Sheet

1. Mandatory Summary 10 points.

Write a summary of three lectures chosen from Lecture 16 to Lecture 20. Note, that any Theorem and Definition, especially those with a name of a renowned mathematician, are worth mentioning. Use the space below. Clearly indicate which lecture you are writing about.

Lecture :

2. Question (simple computation) 10 points.

Find all zeros and singularities of the following functions and classify those (isolated or not, essential, removable or the order of the poles and zeros). Give an explanation of your classification. Moreover compute the residues at all singularities where the residue is defined (removable singularities are ex-cluded). You are allowed to use the fact that all zeros of sin(z) in the complex plane lie at z = πn with n ∈ Z. Simplify your results as much as possible (fractions and factors of π remain as they are)!)

(a) (you must make use of Landau symbols when computing the residue),

(b) (you must make use of the limit formula when computing the residues).

3. Question (simple proof) 10 points.

Consider the real function

(a) Compute the Taylor series at any x0 ∈ R+ \ {1} and show that its radius of convergence is R(x0) = |1 − x0|. When computing the radius of convergence make use of the theorems and corollaries of Lectures 12 & 13 (say which you use and why you can apply those!) so that no ϵ − N criterion is required.

(b) Prove that the union of the open discs of convergence D(x0, |1 − x0|) for x0 ∈ R+ \ {1} of these Taylor series is equal to the set

4. Question (advanced computation) 5 points.

Compute the Laurent series at any point z0 ∈ C and any open annulus centred at z0 of the function

Do not forget to state what the inner and outer radii of convergence Ri < Ro are? (You only need to give a simple explanation via holomorphicity of the function why these are the radii.)

Hint: it is sometimes helpful to make use of the contour integral formula to find the Laurent coefficients! The total number of structurally different Laurent series you should find is 3. Moreover, you will need at some point the product rule for the nth derivative which is

5. Question (advanced proof) 5 points.

Prove the following statement with the help of the theorems and statements up to Lecture 15: Let f(z) and g(z) be two entire functions satisfying |f(z)| ≤ |g(z)| for all z ∈ C, and there exists a z1 ∈ C such that f(z1) = g(z1) ≠ 0. Prove that f(z) = g(z) for all z ∈ C.