Complex Analysis
Problem Set 1
January 27, 2026
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1. Show that f (z) = |z| = x2 + y 2 has a derivative only at the origin.
2. Prove that composition of differentiable functions is differentiable.
3. Let f (z) = ez , describe the image under f of the vertical and the horizontal lines i.e. what are the sets
f (a + it) and f (t + ib), where a and b are constants and t runs through all real numbers?
4. Suppose f : G → C is analytic and G is a connected domain. Show that if f (z) is real for all z ∈ G, then
f is constant.
5. Let G be a region and define G∗ = {z ∈ G : z̄ ∈ G}. If f : G → C is analytic, then prove that
f ∗ : G∗ → C, defined by f ∗ (z) = f (z̄) is also analytic.
6. Is the function z̄z continuous at 0? Why or why not? Is the function zz̄ analytic where it is defined? Why or
why not?
7. Assume that f is analytic in a region and that at every point, either f = 0 or f 0 = 0. Show that f is constant.
8. Describe the following sets:
1. {z : ez = i}
2. {z : ez = −1}
3. {z : cos z = 0}.
sin z
9. Define tan z = cos
z ; where is this function defined and analytic?
10. Let f = u+iv be analytic. Recall that the Jacobian is the function given by the determinant of the following
matrix:
"
#
∂u
∂u
∂(u, v)
∂x
∂y
= ∂v ∂v .
∂(x, y)
∂x
∂y
2
Using Cauchy-Riemann equation, show that this is the same as |f 0 (z)| .
11. Find an example of a function in real variable that is differentiable but not twice differentiable.
12. Express the principal branch of logarithm as a power series with the centre at 1.
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