March 23-27, 2009 Week #9 Practice Quiz Name:
Math 412
Problem 1. (6 points) Find all values of zsuch that exp(5z) = −i.
Problem 2. (6 points) Let α=3π
4. Sketch the image under the function f(z) = logα(z) of the
semidisk x2+y2≤e6with x < 0.
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Math 412 Week #9 Practice Quiz Solutions - Prof. Scott Annin, Quizzes of Mathematics
The solutions to problem 1, 2, and 3 from the week #9 practice quiz of the math 412 course. Problem 1 asks to find all values of z such that exp(5z) = -i. In problem 2, students are asked to sketch the image under the function f(z) = logα(z) of the semidisk x² + y² ≤ e⁶ with x < 0. Lastly, problem 3 asks to prove that the formula log(z₁z₂) = log(z₁) + log(z₂) holds if and only if arg(z₁) + arg(z₂) is within the range (-π, π].
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Pre 2010
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March 23-27, 2009 Week #9 Practice Quiz Name:
Math 412
Problem 1. (6 points) Find all values of z such that exp(5z) = −i.
Problem 2. (6 points) Let α = 34 π. Sketch the image under the function f (z) = logα(z) of the semidisk x^2 + y^2 ≤ e^6 with x < 0.
Problem 3. (8 points) Let z 1 , z 2 ∈ C∗. Prove that the formula Log(z 1 z 2 ) = Log(z 1 ) + Log(z 2 ) holds if and only if −π < Arg(z 1 ) + Arg(z 2 ) ≤ π.