Math 132 Midterm 1 practice
数学期中练习代做 Directions. This is not an assignment to be turned in. These questions are meant to provide practice for Midterm 1.
Directions. This is not an assignment to be turned in. These questions are meant to provide practice for Midterm 1.
Note that these problems are not meant to give a comprehensive review of the material.
Problem 1.
Let z0 = 1 + i. Write e−πi/2z0 in the form x + iy. Sketch both z0 and e−πi/2z0 on the complex plane.
Problem 2. 数学期中练习代做
Let θ0 = π/4. Define the function f : C → C, f(z) = 2eiθ0z.
- Describe the set obtained by applying f to every point in the ray
R = {z ∈ C : Arg z = 3π/4}.
- Describe the set obtained by applying f to every point in the circle
C = {z ∈ C : |z| = 4}.
Problem 3. 数学期中练习代做
(a) Compute all the values of log(−1 + i).
(b) Find a branch F(z) of the multi-valued function f(z) = log(z + i) which satisfies the following conditions:
- F(z) is defined and continuous at all complex numbers except for those z0of the form z0= x − i, x ≥ 0.
- F(−1 − i) = 3πi.
Justify that your branch satisfies these conditions.
Problem 4. 数学期中练习代做
For each multi-valued function below, determine whether ∞ is a branch point for that function.
(a) (z3 − 1)1/3 .
(b) (z5 − 1)1/2 .
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