1.
Use the definition of limit to show for any \(z_0 \in \C\) that \(\displaystyle \lim_{z \to z_0} (a z + b) = a z_0 + b\text{.}\)
Exercises 2.5 Exercises
2.
Evaluate the following limits or explain why they don’t exist.
(a)
\(\displaystyle \lim_{z \to i} \frac{i z^3 - 1}{z + i}\)
(b)
\(\displaystyle \lim_{z \to 1 - i} \left(x + i(2x +
y)\right)\)
3.
Prove that, if a limit exists, then it is unique.
4.
Prove Proposition 2.1.6.
5.
Let \(f: G \to \C\) and suppose \(z_0\) is an accumulation point of \(G\text{.}\) Show that \(\lim_{ z \to z_0 } f(z) = 0\) if and only if \(\lim_{ z \to z_0 } |f(z)| = 0\text{.}\)
6.
Proposition 2.1.4 is useful for showing that limits do not exist, but it is not at all useful for showing that a limit does exist. For example, define
\begin{equation*}
f(z) = \frac{x^2y}{x^4+y^2} \quad \text{ where } \quad z=x+iy\ne0\,\text{.}
\end{equation*}
Show that the limits of \(f\) at \(0\) along all straight lines through the origin exist and are equal, but \(\displaystyle\lim_{z\to0}f(z)\) does not exist.
7.
Suppose that \(f(z) = u(x,y) + i \, v(x,y)\) and \(z_0 =
x_0 + i \, y_0\text{.}\) Prove that
\begin{equation*}
\lim_{ z \to z_0 } f(z) = u_0 + i \, v_0
\end{equation*}
if and only if
\begin{equation*}
\lim_{ (x,y) \to (x_0, y_0) } u(x,y) = u_0 \qquad \text{ and
} \qquad \lim_{ (x,y) \to (x_0, y_0) } v(x,y) = v_0 \, \text{.}
\end{equation*}
8.
9.
Show that the function \(g: \C \to \C\) given by
\begin{equation*}
g(z) = \begin{cases}\frac{\, \overline z \, } z \amp \text{
if } z \ne 0 \, , \\ 1 \amp \text{ if } z = 0 \end{cases}
\end{equation*}
is continuous on \(\C \setminus \{ 0 \}\text{.}\)
10.
Determine where each of the following functions \(f: \C \to
\C\) is continuous:
(a)
\(f(z) = \begin{cases}0 \amp \text{ if } z=0 \text{ or
\(|z|\) is irrational, } \\ \frac 1 q \amp \text{ if }
|z| = \frac p q \in \Q \setminus \{ 0 \} \text{
(written in lowest terms). } \end{cases}\)
(b)
\(f(z) = \begin{cases}0 \amp \text{ if } z=0 , \\ \sin
\phi \amp \text{ if } z = r \, e^{ i \phi } \ne 0
. \end{cases}\)
11.
Show that the two definitions of continuity in Section 2.1 are equivalent. Consider separately the cases where \(z_0\) is an accumulation point of \(G\) and where \(z_0\) is an isolated point of \(G\text{.}\)
12.
Consider the function \(f: \C \setminus \{ 0 \} \to \C\) given by \(f(z) = \frac 1 z\text{.}\) Apply the definition of the derivative to give a direct proof that \(f'(z) = - \frac 1 { z^2 }\text{.}\)
13.
Prove Proposition 2.1.11.
14.
Prove Proposition 2.2.5.
15.
Find the derivative of the function \(T(z) :=
\frac{az+b}{cz+d}\text{,}\) where \(a,b,c,d \in \C\) with \(ad - bc \neq 0\text{.}\) When is \(T'(z) = 0\text{?}\)
16.
Prove that if \(f(z)\) is given by a polynomial in \(z\) then \(f\) is entire. What can you say if \(f(z)\) is given by a polynomial in \(x = \Re z\) and \(y = \Im z\text{?}\)
17.
Prove or find a counterexample: If \(u\) and \(v\) are real valued and continuous, then \(f(z) = u(x,y) + i \, v(x,y)\) is continuous; if \(u\) and \(v\) are (real) differentiable then \(f\) is (complex) differentiable.
18.
Where are the following functions differentiable? Where are they holomorphic? Determine their derivatives at points where they are differentiable.
(a)
\(f(z) = e^{-x}e^{-iy}\)
(b)
\(f(z) = 2 x + i x y^2\)
(c)
\(f(z) = x^2 + i y^2\)
(d)
\(f(z) = e^x e^{-i y}\)
(e)
\(f(z) = \cos x \cosh y - i \sin x \sinh y\)
(f)
\(f(z)=\Im z\)
(g)
\(f(z) = |z|^2 = x^2 + y^2\)
(h)
\(f(z) = z \Im z\)
(i)
\(f(z)= \frac{ ix+1 } y\)
(j)
\(f(z)=4(\Re z)(\Im z)-i(\conj z)^2\)
(k)
\(f(z)=2xy-i(x+y)^2\)
(l)
\(f(z)=z^2-\conj z^2\)
19.
The Jacobian of a transformation \(u=u(x,y)\text{,}\) \(v=v(x,y)\) is the determinant of the matrix
\begin{equation*}
\displaystyle\begin{bmatrix}\fderiv ux\amp \fderiv uy\\ \fderiv vx \amp \fderiv vy \end{bmatrix} \,\text{.}
\end{equation*}
Show that if \(f = u+iv\) is holomorphic then the Jacobian equals \(\abs{f'(z)}^2\text{.}\)
20.
Define \(f(z) = 0\) if \(\Re(z)\cdot\Im(z)=0\text{,}\) and \(f(z)=1\) if \(\Re(z)\cdot\Im(z)\ne0\text{.}\) Show that \(f\) satisfies the Cauchy–Riemann equation (2.2) at \(z = 0\text{,}\) yet \(f\) is not differentiable at \(z = 0\text{.}\) Why doesn’t this contradict Theorem 2.3.1(b)?
21.
Prove: If \(f\) is holomorphic in the region \(G \subseteq \C\) and always real valued, then \(f\) is constant in \(G\text{.}\)
22.
Prove: If \(f(z)\) and \(\overline{ f(z) }\) are both holomorphic in the region \(G \subseteq \C\) then \(f(z)\) is constant in \(G\text{.}\)
23.
Suppose \(f\) is entire and can be written as \(f(z) =
u(x) + i \, v(y)\text{,}\) that is, the real part of \(f\) depends only on \(x = \Re(z)\) and the imaginary part of \(f\) depends only on \(y = \Im(z)\text{.}\) Prove that \(f(z) = az + b\) for some \(a \in \R\) and \(b \in \C\text{.}\)
24.
Suppose \(f\) is entire, with real and imaginary parts \(u\) and \(v\) satisfying \(u(x,y) \, v(x,y)=3\) for all \(z=x+iy\text{.}\) Show that \(f\) is constant.
25.
Prove that the Cauchy–Riemann equations take on the following form in polar coordinates:
\begin{equation*}
\frac{ \partial u }{ \partial r } \ = \ \frac 1 r \, \frac{
\partial v }{ \partial \phi } \qquad \text{ and } \qquad
\frac 1 r \, \frac{ \partial u }{ \partial \phi } \ = \ -
\frac{ \partial v }{ \partial r } \, \text{.}
\end{equation*}
26.
For each of the following functions \(u\text{,}\) find a function \(v\) such that \(u+iv\) is holomorphic in some region. Maximize that region.
(a)
\(u(x,y) = x^2 - y^2\)
(b)
\(u(x,y) = \cosh (y) \sin (x)\)
(c)
\(u(x,y) = 2x^2 + x + 1 - 2y^2\)
(d)
\(u(x,y) = \frac{x}{x^2 + y^2}\)
27.
Is \(u(x,y) = \frac x {x^2+y^2}\) harmonic on \(\C\text{?}\) What about \(u(x,y) = \frac{ x^2 }{ x^2+y^2 }\text{?}\)
28.
Consider the general real homogeneous quadratic function \(u(x,y) = ax^2+bxy+cy^2\,\text{,}\) where \(a\text{,}\) \(b\) and \(c\) are real constants.
(a)
(b)
If \(u\) is harmonic then show that it is the real part of a function of the form \(f(z) = Az^2\) for some \(A \in \C\text{.}\) Give a formula for \(A\) in terms of \(a\text{,}\) \(b\) and \(c\text{.}\)
29.
30.
Prove that, If \(G\subseteq\C\) is a region and \(f: G \to \C\) is a complex-valued function with \(f''(z)\) defined and equal to \(0\) for all \(z \in
G\text{,}\) then \(f(z) = a z + b\) for some \(a,
b \in \C\text{.}\)
Hint.
Use Theorem 2.4.2 to show that \(f'(z) = a\text{,}\) and then use Theorem 2.4.2 again for the function \(f(z) - a z\text{.}\)
