MATH 417 MIDTERM 1
This midterm is due Wednesday March 5 in the beginning of class. You may use
your class notes and the course text book. You may not use any other materials,
including other text books, the web, question centers, etc. The work should be
yours and yours alone. Please do not collaborate. There are 10 problems each
worth 10 points.
Problem 1 Find all solutions of the equation z8=โ1 + โ3i. Calculate all the
values of (โ1+ โ3i)1/8using the definition of roots via logarithms. Show that your
two answers are the same.
Problem 2 Find the principal values of the following expressions.
(1) (2 โ2i)(1+i)
(2) (2 โ2โ3i)(3+4i)
Problem 3 Determine whether the function eโxcos y+x4โ6x2y2+y4can be the
real part of an analytic function. If so, find all analytic functions that have it as
their real parts.
Problem 4 Let f(z) = u(x, y) + iv (x, y) be an analytic function in a domain D.
Let a, b be two real numbers. Assume that z0=x0+iy0is on the level curves
u(x, y) = aand v(x, y) = band that f0(z0)6= 0. Show that the tangent lines to the
curves u(x, y) = aand v(x, y) = bat z0are perpendicular. (Hint: Show that the
dot product of the gradient vectors are zero.)
Problem 5 Calculate the integral
ZC
(ez2+z2)dz
where Cis the positively oriented boundary of a rectangle with vertices โ1โi, 2โ
i, 2 + i, โ1 + i.
Problem 6 Estimate the integral
ZCR
2z2+ 7
z6+ 2z3+ 1dz,
where CRis the circle |z|=R(assume R >> 0) positively oriented. Show that the
integral tends to zero as Rtends to infinity.
Problem 7 Calculate the integral
ZC
ezcos(z)
z3dz,
where Cis the unit circle taken with the positive orientation
1
