MAT 461 - Some Review For Final
Posted December 4
These problems are meant to help supplement your studying. They will cover many, but not all of the topics
that may appear on the final.
1. Sketch the sets of points determined by the following conditions. Are these regions (open, closed or
neither), (bounded or unbounded), (connected or not connected)?
• |z+ 2 −i|= 2
• |z−3i| ≤ 3
• |z−1|=|z−i|
2. Assuming that |z|= 1, show that
2z+ 1
z2+ 4iz
≤1
3. Let z= 2 −2i.
•Put zin exponential form.
•Find z4.
•Find the roots z1
2. Give the roots in principal argument form. Which is the principal root?
•Find Log z(Log zhas branch cut at −π).
•Simplify 1
¯zinto x+iy form.
4. Let f(z) = 3z2−1
z(z−i). Compute the following limits:
•limz→0f(z)
•limz→1f(z)
•limz→∞ f(z)
5. Use the Cauchy-Riemann equations to check whether the following are analytic. (f(z) = u(x, y) +
iv(x, y)).
•ln(x2+y2) + i2 tan−1(y
x)
•x
x2+y2+iy
x2+y2
6. Find a harmonic conjugate for x2+ 2x−y2.
7. Find the derivatives of the following (assume an appropriate branch cut when necessary)
•(1 + i)z2
•log( 1
z)
•cos−1(ez)
