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MATH 4171 Functions of Complex Variable

Summer 2023

Midterm Exam

Please read the following requirements  carefully:

1.  Steps/reasoning are required to support your work.  Answers without steps/reasoning are not acceptable.

2.  You can only use the theorems/definitions/lemmas covered in Sections 1-42 when you refer to them in your solutions.

1. [35pts] Prove that |z1 +z2 | 2 +|z1  − z2 | 2  = 2(|z1 | 2 +|z2 | 2 ) for any complex numbers z1  and z2 .

2. [35pts] Rewrite (1 − i√3)20  to the form a + bi.  (The answer in scientific notation is fine.)

3. [35pts] Find all of the roots of (−64)1/6  in rectangular coordinates.

4. [35pts] Find all differentiable points and all singular points of the functions

(a) f(z) = 2xy + i(x2 + y2 )       (b) f(z) = zRez

5. [35pts] Evaluate

(a) log i2/1 (b)（ 1i）1+i

6. [45pts] Solve the equation sinhz = 2i.

7. [Bonus 10pts] If z + z − 1  = 2cosθ, show that zn  + z −n  = 2cos(nθ).