Midterm 1 Solutions

This is false. The principal argument always has to be in the range from to , and the sum of two numbers in this range is not necessarily also in that range.

Question 1(b)

This is false. In fact, if we write z=x+iy and w=u+iv, then (x+iy)(u+iv) = (xu-yv) + i(xv+yu), so we see that the real part Re(zw) equals xu-yv = Re(z)Re(w) - Im(z)Im(w). This will not in general be equal to Re(z)Re(w), unless the product Im(z)Im(w) happens to be zero.

Question 1(c)

Question 1(d)

Question 1(e)

Question 2

Question 3(a)

Question 3(b)

Question 4

Question 5

Question 6

Question 7(a)

Question 7(b)

In order for f to be analytic at a point z, it must be differentiable on some disk centred at z. But we know f is differentiable only on a single line. There is no disk (of non-zero radius) that can be completely contained within that line. Therefore, there is no disk on which f is differentiable, and hence there is no point at which f is analytic.

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