MAT 461 Applied Complex Analysis /100
October 8, 1999 Test 1 name
Justify all major steps that involve substantial complex
analysis reasoning. On the other hand, there is no need
for lots of detail in steps that involve only calculus or
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algebra – often a computer print-out may be adequate documentation. You may use MAPLE throughout,
but it is YOUR responsibility to demonstrate that you have mastered the new material of this class.
1. a. Write the number (1 + √3)4in the form a+bi with aand breal.
b. Find all solutions of the equation z3=i.
Bonus: Find all solutions of the equation zi=i.
2. a. Describe the set A={z∈C: 2 ≤ <(z)≤3 and 0 ≤ =(z)≤π}in words, and sketch it.
b. Describe the image f(A) of the set Aunder the map f:z7→ ez.
Carefully explain why fmaps onto the set you claim to be f(A).
c. Show that fmaps the region Aone-to-one onto the set f(A) you found in part b.
3.a. Show that if fis analytic, then its real part u=<(f) is harmonic.
You may assume (we will prove this soon) that the real and imaginary
parts of every analytic function have continuous derivatives of all orders.
b. Show that if uand vare conjugate harmonic functions, then the product uv is harmonic .
Hint: For an elegant argument consider =((u+iv)2).
Bonus: Give an explicit counterexample to show that the product of two harmonic
functions need not be harmonic.
c. Demonstrate that u: (x, y )7→ log √x2+y2is harmonic, and find all its harmonic conjugates.
4.a. Find the Jacobian matrix of partial derivatives of F:R27→ R2defined by F(x, y) = (x2,−y2).
b. Explain in elementary terms (e.g. using the definition of differentiability) why the function
f:C7→ Cdefined by f(z) = (<(z))2−i(=(z))2is not analytic, even though Fis (real) differentiable.
c. Show that if a function mapping Cto Cis (complex) differentiable at a point z0then its real
and imaginary parts satisfy the Cauchy-Riemann equations at z0.
d. Find all points zat which the function fof part a. satisfies the Cauchy-Riemann equations.
e. Find all points where the function fis analytic.
5.a. Use the definition of the complex cosine (in terms of the complex exponential) to derive the
expression −ilog(z+√z2−1) for cos−1(z) in terms of the complex logarithm.
b. Find all values of cos i.
