Math 303 - Complex Variables
Homework due February 20
Recall that in class we defined the complex logarithm by the multi-valued function
log z= log |z|+iarg z.
Of course, this function outputs a set since arg zis the set of angles that zmakes with the positive
real axis.
We also defined the principal logarithm, which will be a single-valued function, by using the principal
argument Arg zinstead of arg z:
Log (z) = log |z|+iArg z
Question 1. Evaluate each of the following. Be sure to note when you are asked to find log zor Log z
(thus you at times are asked for a set as an answer and at times a single number as an answer).
(a) log i
(b) log(1 โi)
(c) Log (โi)
(d) Log (โ3 + i)
Question 2. Consider the principal logarithm
Log (z) = log |z|+iArg z.
(a) What is the domain of Log z? [Hint: What is the range of ez?]
(b) What is the range of Log z?
Question 3. Below, we shall prove that the Cauchy-Riemann equations hold for Log z. Recall that
Log z= log |z|+iArg z.
Thus, in terms of u+iv,
u= log |z|and v= Arg z.
(a) Write u= log |z|in terms of xand yby plugging substituting z=x+iy.
(b) Use (a) to find the partial derivatives โ
โx and โ
โy of log |z|.
(c) Use trigonometry to show that if z=x+iy, then
Arg z= arctan ๎y
x๎.
(d) Use (c) to find the partial derivatives โ
โx and โ
โy of Arg z.
(e) Use the partial derivatives you found in (b) and (c) to verify that the Cauchy-Riemann equations
hold.
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