Complex Variables – Exercises

1 Complex numbers

Exercise 1.1 Find real numbers x, y such that

3x + 2iy − ix + 5y = 7 + 5i.

Exercise 1.2 Prove that

|Re z| + |Im z| ≤ \sqrt{2}|z|, z ∈ ℂ.  (1.1)

Study the case of equality in (1.1).

Solution: With z = x + iy, we have to prove that

|x| + |y| ≤ \sqrt{2}\sqrt{x^{2} + y^{2}}.  (1.2)

Squaring both sides of (1.2), we need to prove that

x^{2} + y^{2} + 2|xy| ≤ 2(x^{2} + y^{2}),

which is equivalent to

x^{2} − 2|xy| + y^{2} ≥ 0.

This last inequality holds trivially since x^{2} − 2|xy| + y^{2} = (|x| − |y|)^{2}. We also conclude that there will be equality if and only if x = y, i.e. if and only if z is of the form z = x(1 + i) for some real number x.

Inequality (1.2) is a special case of the Cauchy–Schwartz inequality in ℝ

|x_{1}x_{2} + y_{1}y_{2}| ≤ \sqrt{x_{1}^{2} + y_{1}^{2}}\sqrt{x_{2}^{2} + y_{2}^{2}} ,  (1.3)

obtained by taking x_{1} = x, y_{1} = y and x_{2} = y_{2} = 1. As known from calculus, there is equality in (1.3) if and only if there is a real number λ such that

( \begin{matrix}x_{1} \\ y_{1} \end{matrix} ) = λ ( \begin{matrix}x_{2} \\ y_{2} \end{matrix} ) .

This leads to x = λ and y = λ, and we get again the fact that there is equality in (1.2) if and only if z is of the form z = x(1 + i) for some real number x. □

Exercise 1.3 Draw the set of points in the complex plane satisfying:

a) |z − 1 + i| = 1.

b) z^{2} + \overline{z}^{2} = 2.

c) |z − i| = |z + i|. Exercise 1.4 Find all solutions of the equation

z^{8} = \frac{1 + i}{\sqrt{3} − i} .

Exercise 1.5 Show that

(\frac{1 + i tan α}{1 − i tan α})^{n} = \frac{1 + i tan(nα)}{1 − i tan(nα)} .

Exercise 1.6 Prove the following equalities and inequalities for any complex numbers:

| |z| − |w| | ≤ |z ± w| ≤ |z| + |w|  (1.4)
|z + w|^{2} + |z − w|^{2} = 2(|z|^{2} + |w|^{2})  (1.5)
|1 + z\overline{w}|^{2} + |z − w|^{2} = (1 + |z|^{2})(1 + |w|^{2})  (1.6)
|1 + z\overline{w}|^{2} − |z − w|^{2} = (1 − |z|^{2})(1 − |w|^{2})  (1.7)
| ∑_{1}^{p} z_{j}w_{j}|^{2} ≤ (∑_{1}^{p} |z_{j}|^{2})(∑_{1}^{p} |w_{j}|^{2}).  (1.8)

Exercise 1.7 Prove that for any complex number z different from 1 we have

1 + z + ⋯ + z^{n} = \frac{1 − z^{n+1}}{1 − z} .

Exercise 1.8 Using the preceding exercise, prove that, for all real θ such that sin \frac{θ}{2} ≠ 0,

1 + cos θ + ⋯ + cos(nθ) = \frac{1}{2} + \frac{sin ((n + \frac{1}{2}) θ))}{2 sin \frac{θ}{2}} .

The next exercise extends Exercise 1.8. Exercise 1.9 Show that

∑_{k=0}^{n−1} cos(a + kb) = \frac{sin(\frac{nb}{2})}{sin \frac{b}{2}} · cos(a + (n − 1)\frac{b}{2})  (1.9)
∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) cos(a + kb) = 2^{n} cos^{n} \frac{b}{2} cos(a + \frac{nb}{2}).  (1.10)

Solution: For the first formula, we compute

(∑_{k=0}^{n−1} cos(a + kb)) + i (∑_{k=0}^{n−1} sin(a + kb)) = ∑_{k=0}^{n−1}(cos(a + kb) + sin(a + kb))
= ∑_{k=0}^{n−1} e^{i(a+kb)}
= e^{ia} \frac{1 − e^{inb}}{1 − e^{ib}}
= e^{ia} \frac{e^{i \frac{nb}{2}}}{e^{i \frac{b}{2}}} \frac{e^{−i \frac{nb}{2}} − e^{i \frac{nb}{2}}}{e^{−i \frac{b}{2}} − e^{i \frac{b}{2}}}
= e^{i(a+ \frac{nb}{2} − \frac{b}{2} )} \begin{matrix}sin & \frac{nb}{2} \\ sin & \frac{b}{2} \end{matrix}

To prove the second formula, we first recall Newton’s binomial formula:

(α + β)^{n} = ∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) α^{k}β^{n−k},  (1.11)

which for β = 1 leads to

(1 + α)^{n} = ∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) α^{k}.

To compute the sum (1.10) we compute

(∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) cos(a + kb)) + i (∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) sin(a + kb)) = ∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) e^{i(a+kb)}
= e^{ia} (∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) e^{ikb})
= e^{ia}(1 + e^{ib})^{n}
= e^{ia}e^{i \frac{b}{2}} (e^{−i \frac{b}{2}} + e^{i \frac{b}{2}} )^{n}
= e^{i(a+ \frac{b}{2} )}(2 cos \frac{b}{2})^{n}
= 2^{n} (cos \frac{b}{2})^{n} e^{i(a+ \frac{b}{2} )},

and hence the result. □

More generally, one has for a, b, r ∈ ℝ and n a positive integer

∑_{k=0}^{n−1} r^{k} cos(a + kb) = \frac{cos a − r cos(a − b) − r^{n} cos(a + nb) + r^{n+1} cos(a + (n − 1)b)}{1 + r^{2} − 2r cos b}
∑_{k=0}^{n−1} r^{k} sin(a + kb) = \frac{sin a − r sin(a − b) − r^{n} sin(a + nb) + r^{n+1} sin(a + (n − 1)b)}{1 + r^{2} − 2r cos b}

Exercise 1.10 Compute, for a real number r with r < 1, the sums

∑_{n=0}^{∞} r^{n} sin(nθ)

and

∑_{n=0}^{∞} r^{n} cos(nθ).

In a similar vein, try the following:

∑_{n=1}^{∞} nr^{n} sin(nt) = \frac{r(1 − r^{2}) sin t}{(1 − 2r cos t + r^{2})^{2}} .  (1.12)

Hint: First prove that:

z(1 + 2z + 3z^{2} + 4z^{3} + ⋯) = \frac{z}{(1 − z)^{2}}  (1.13)

Exercise 1.11 Prove that for all z ∈ ℂ,

|\sqrt{z^{2} − 1} + z| + |\sqrt{z^{2} − 1} − z| = |z − 1| + |z + 1|.

Solution: Taking squares, the equality to be proved is equivalent to

|\sqrt{z^{2} − 1} + z|^{2} + |\sqrt{z^{2} − 1} − z|^{2} + 2|\sqrt{z^{2} − 1} + z| · |\sqrt{z^{2} − 1} − z| =  (1.14)

= |z − 1|^{2} + |z + 1|^{2} + 2|z − 1| · |z + 1|.

Taking into account the equality

|ω_{1} + ω_{2}|^{2} + |ω_{1} − ω_{2}|^{2} = 2(|ω_{1}|^{2} + |ω_{2}|^{2}), ω_{1}, ω_{2} ∈ ℂ, (1.15)

we have

|z − 1|^{2} + |z + 1|^{2} = 2(|z|^{2} + 1)  (1.16)
|\sqrt{z^{2} − 1} + z|^{2} + |\sqrt{z^{2} − 1} − z|^{2} = 2(|z^{2} − 1| + |z^{2}|).  (1.17)

(For the first equality, take ω_{1} = z and ω_{2} = 1; for the second, take ω_{1} = \sqrt{z^{2} − 1} and ω_{2} = z.)

Furthermore, it is clear that

|\sqrt{z^{2} − 1} + z| · |\sqrt{z^{2} − 1} − z| = |(z^{2} − 1) − z^{2}| = 1  (1.18)
|z − 1| · |z + 1| = |z^{2} − 1|.  (1.19)

Therefore, (1.14) is equivalent to

2(|z^{2} − 1| + |z^{2}|) + 2 = 2(|z^{2}| + 1) + 2|z^{2} − 1|,

which trivially holds. □

Exercise 1.12 Which of the following sets are open? closed?

(a) {z ∈ ℂ : |z − (3 − 5i)| < 2} .

(b) {z ∈ ℂ : |z − (3 − 5i)| ≤ 2, |z − (4 − 7i)| < 2} .

(c) {z ∈ ℂ : |Re z| < |z|} .

(d) {z ∈ ℂ : |Re z| ≤ |z|/2} .

(e) {z ∈ ℂ : |Re z| ≥ |z|/2, |Im z| ≥ |z|/2} .

(f) {(l + mi)/2^{n} : l, m, n = 0, 1, 2 … } . Exercise 1.13 Find the boundary and the closure of each of the sets defined in the preceding exercise. Exercise 1.14 Find all accumulation points of the set {z_{n} : n ∈ ℕ} for each of the following sequences (z_{n})_{n=1}^{∞}:

(a) z_{n} = \frac{3 − 2i}{n^{2}}.

(b) z_{n} = (\frac{1 + i}{2})^{n}.

(c) z_{n} = (\frac{1 + i\sqrt{3}}{2})^{n}.

(d) z_{n} = (\frac{4 + 3i}{5})^{n}. (Provide a heuristic argument, but do not try to prove your claim.)

2 Limits and Continuity

Exercise 2.1 Express the following functions in the form f(z) = u(x, y) + iv(x, y):

(a) f(z) = z^{4} + (5 − i)z^{2} − 6iz.

(b) f(z) = \frac{z^{2} + i}{z − 1}.

(c) f(z) = z^{2} − |z|^{2}.

(d) f(z) = \frac{\overline{z}}{z + \overline{z} + 1}.

Exercise 2.2 Let f(z) = 3x^{2} + y^{3} − 2xy + i(x^{3} − 2xy^{2} + 3y), where z = x + iy. Express f(z) in terms of z. Exercise 2.3 Prove directly from the definition of limit that:

(a) lim_{z→3−i} z^{2} = 10 − 6i.

(b) lim_{z→z_{0}} Im z = Im z_{0}.

(c) lim_{z→0} \frac{\overline{z}^{3}}{z} = 0.

Exercise 2.4 Find the following limits:

(a) lim_{z→−2+i}(1 − i)z^{2} − 4iz.

(b) lim_{z→3−5i} \frac{iz − 1}{z − 3i}.

(c) lim_{z→3+4i} \frac{z^{2} − 6z + 25}{z^{2} − 8z + (−1 + 32i)}.

(d) lim_{z→1+i\sqrt{3}} \frac{z^{100} + \overline{z}^{100}}{z^{2} − \overline{z}^{2}} .

(e) lim_{z→1+i\sqrt{3}} \frac{z^{300} − \overline{z}^{300}}{z^{48} − \overline{z}^{48}} .

(f) lim_{z→0} \frac{sin(z + \overline{z}) · (e^{|z|} − 1)}{z}.

Exercise 2.5 Consider the function f : ℂ − {0} ⟶ ℂ − {0} defined by:

f(z) = \frac{x^{2} − y^{2}}{x^{2} + y^{2}} + i\frac{2xy}{x^{2} + y^{2}} , z ≠ 0 .

(a) Prove that f is continuous throughout its domain of definition.

(b) Can you extend f to the whole of ℂ while keeping it continuous?

(c) Explain the geometric action of f, and interpret your answers to the preceding parts in view of that.

Exercise 2.6 Prove that the functions f(z) = Re z and g(z) = Im z are continuous everywhere.

3 Differentiable and Analytic functions

Exercise 3.1 Prove directly from the definition that each of the following functions is nowhere differentiable in the complex plane:

(a) f(z) = |z|.

(b) f(z) = Re z.

(c) f(z) = Im z.

(d) f(z) = \overline{z}. Exercise 3.2 Find at which points the following functions are differentiable, and calculate their derivatives there:

(a) f(z) = \frac{iz − 2}{z^{2} + 1}.

(b) f(z) = \{ \begin{matrix}\frac{z^{2} − (11 + 4i)z + (27 + 23i)}{z^{2} − 12z + 37}, & z ≠ 6 ± i , \\ \frac{−4 + 2i}{5}, & z = 6 + i , \\ \frac{−4 − 2i}{5}, & z = 6 − i . \end{matrix}

(c) f(z) = |z|^{2}.

(d) f(z) = |z|^{2}|z^{2} + 1|^{2}. Exercise 3.3 Let f and g vanish and be differentiable at the point z_{0}, with g′(z_{0}) ≠ 0. Prove that:

lim_{z→z_{0}} \frac{f(z)}{g(z)} = \frac{f′(z_{0})}{g′(z_{0})} .

Exercise 3.4 Show that the function f defined by

f(z) = \{ \begin{matrix}\frac{\overline{z}^{2}}{z} , & z ≠ 0 , \\ 0, & z = 0 . \end{matrix}

is not differentiable at the origin, but the Cauchy–Riemann equations hold there.

Solution: The continuity of f at the origin follows from:

|f(z) − f(0)| = f(z) = \frac{|\overline{z}^{2}|}{|z|} = |z| ⟶_{z→0} 0 .

We now consider

\frac{f(z) − f(0)}{z} = \frac{\overline{z}^{2}}{z^{2}} .

Let z = x be real. For such z we have \frac{f(z)−f(0)}{z} = \frac{\overline{x}^{2}}{x^{2}} = 1, and therefore

lim_{z ⟶ 0} \frac{f(z) − f(0)}{z} = 1.  (3.1)

Take now z = (1 + i)y with y real. Then

\frac{f(z) − f(0)}{z} = \frac{\overline{(1 + i)y}^{2}}{((1 + i)y)^{2}} = \frac{\overline{(1 + i)^{2}}}{(1 + i)^{2}} = −1,

and so

lim_{z=(1+i)y→0} \frac{f(z) − f(0)}{z} = −1.  (3.2)

Comparing (3.1) and (3.2), we get that f is not differentiable at the origin. Let z = x + iy. We have f(z) = u(x, y) + iv(x, y) with

u(x, y) = \frac{x^{3} − 3xy^{2}}{x^{2} + y^{2}} , v(x, y) = \frac{−3x^{2}y + y^{3}}{x^{2} + y^{2}}

for z ≠ 0 and u(0, 0) = v(0, 0) = 0. To show that the Cauchy–Riemann equations hold at the origin, we compute the various partial derivatives of the real and imaginary parts of f. We have

u_{x}(0, 0) = lim_{x ⟶ 0} \frac{u(x, 0) − u(0, 0)}{x} = lim_{x ⟶ 0} \frac{\frac{x^{3}}{x^{2}}}{x} = lim_{x→0} \frac{x^{3}}{x^{3}} = 1.

Similarly,

v_{y}(0, 0) = lim_{y→0} \frac{v(0, y) − v(0, 0)}{y} = lim_{y→0} \frac{\frac{y^{3}}{y^{2}}}{y} = lim_{y→0} \frac{y^{3}}{y^{3}} = 1.

So the Cauchy–Riemann equation u_{x}(0, 0) = v_{y}(0, 0) holds. The other Cauchy–Riemann equation is proved in a similar way. □

Exercise 3.5 Use the necessity of the Cauchy–Riemann equations to show that the following functions are nowhere differentiable, or are differentiable at most at “very few” points:

(a) f(z) = 5x^{2} − (3 + i)y^{2}.

(b) f(z) = z + \overline{z}.

(c) f(z) = e^{x}e^{−iy}.

Exercise 3.6 Prove that if the real and imaginary parts of two functions f(z) and g(z) satisfy the Cauchy–Riemann equations, then the same holds for:

(a) h(z) = f(z) + g(z).

(b) h(z) = f(z)g(z).

Exercise 3.7 Show directly that the Cauchy–Riemann equations are satisfied if f(z) = z^{n} where n is a positive integer.

Exercise 3.8 Show that the following functions are entire:

(a) f(z) = 3x + y + i(3y − x).

(b) f(z) = sin x cosh y + i cos x sinh y.

Solution: For case (a), one can check that the Cauchy–Riemann equations hold at all points (do it!). It is quicker to note that

f(z) = 3x + 3iy − i(x + iy) = (3 − i)z.

For (b) we verify that the Cauchy–Riemann equations hold throughout ℂ. In fact

u_{x}(x, y) = cos x cosh y,
u_{y}(x, y) = sin x sinh y
v_{x}(x, y) = − sin x sinh y
v_{y}(x, y) = cos x cosh y,

and hence for all z = x + iy:

u_{x}(x, y) = v_{y}(x, y), u_{y}(x, y) = −v_{x}(x, y).
□

Exercise 3.9 Find the singularities of the functions

(a) f(z) = \frac{2z + 1}{z(z^{2} + 1)}.

(b) f(z) = \frac{z^{2} + i}{z^{2} − 3z + 2}.

(c) f(z) = \frac{1}{(z + 2)(z^{2} + 2z + 2)}.

(d) f(z) = \frac{z − (1 + i)}{z^{2} − (4 + 3i)z + (1 + 5i)}.

Solution: In the three examples, the function f is a quotient of polynomials. Since polynomials are entire functions, in each case the functions is analytic at all points where the denominator does not vanish. So:

Case (a): f is analytic in ℂ − {0, i, −i}.

Case (b): f is analytic in ℂ − {1, 2}.

Case (c): f is analytic in ℂ − {−2, −1 + i, −1 − i}. For this last case, note that the roots of z^{2} + 2z + 2 = 0 are −1 + i and −1 − i.

Case (d): The roots of the denominator are z_{1} = 3+2i and z_{2} = 1+i, and hence the function is analytic in ℂ − {z_{1}, z_{2}}. Note that the denominator can be written as (z − z_{1})(z − z_{2}). Thus:

f(z) = \frac{(z − z_{2})}{(z − z_{1})(z − z_{2})} = \frac{1}{z − z_{1}} .

The function g(z) = \frac{1}{(z−z_{1})} is analytic in ℂ − \{z_{1}\}, and is an analytic extnesion of g. The point z_{2} = 1 + i is a removable singularity of f. □

Exercise 3.10 Let f be an analytic function in some open connected set S. Prove that each of the following properties of f implies that it is constant throughout S:

(a) f(z) is real for every z ∈ S.

(b) f assumes only values on the main diagonal {x + ix : x ∈ ℝ}.

(c) \overline{f(z)} is analytic throughout S.

(d) |f(z)| is constant throughout S.

(e) f assumes only values on the parabola {x + ix^{2} : x ∈ ℝ}.

(f) f assumes only values on the hyperbola {x + i/x : x ∈ ℝ − {0}}.

4 Elementary Functions

Exercise 4.1 Calculate the following:

(a) e^{−3+5πi}.

(b) e^{\frac{−2+5πi}{4}} .

(c) e^{\frac{1+13πi}{3}} . Exercise 4.2 Find all solutions of the following equations:

(a) e^{z^{2}} = e^{z}.

(b) e^{z^{2}} = (e^{z})^{2}.

(c) e^{2e^{z}} = e^{e^{z}}.

(d) \overline{e^{iz}} = e^{i\overline{z}}. Exercise 4.3 Let the function f(z) = u(x, y) + iv(x, y) be analytic in some open set S. Prove that the functions U and V defined by

U(x, y) = e^{u(x,y)} cos v(x, y), V (x, y) = e^{u(x,y)} sin v(x, y),

satisfy the Cauchy-Riemann equations without actually calculating any partial derivatives. Exercise 4.4 Let A be the image of the line {x + ix : x ∈ ℝ} under the exponential map. Write the equation of A in polar coordinates. Exercise 4.5 Prove the following identities for any complex numbers z = x + iy, z_{1}, z_{2}:

(a) cos(z_{1} + z_{2}) = cos z_{1} cos z_{2} − sin z_{1} sin z_{2}.

(b) sin^{2} z + cos^{2} z = 1.

(c) sin (\frac{π}{2} − z) = cos z.

(d) | sin z|^{2} = sin^{2} x + sinh^{2} y.

(e) \overline{cos iz} = cos i\overline{z}.

(f) sin^{3} z = \frac{3}{4} sin z − \frac{1}{4} sin 3z . Exercise 4.6 Prove the following inequalities for any z = x + iy:

(a) | cos z| ≥ | cos x|.

(b) | sinh y| ≤ | cos z| ≤ | cosh y|.

(c) | sin z| ≤ sinh |z|.

Exercise 4.7 For which values of a does the equation sin z − cos z = a have a solution? How many solutions does it have for such values of a? Exercise 4.8 Find all solutions of the following equations:

(a) \overline{sin iz} = sin i\overline{z}.

(b) sin z_{1} = sin z_{2}.

(c) sin z = 3.

(d) sin z + \sqrt{3} cos z = −4. Exercise 4.9 Prove the following identities:

(a) cosh(z_{1} + z_{2}) = cosh z_{1} cosh z_{2} + sinh z_{1} sinh z_{2}.

(b) cosh^{2} z − sinh^{2} z = 1.

(c) | cosh z|^{2} = sinh^{2} x + cos^{2} y. Exercise 4.10 Find all solutions of the following equations:

(a) cosh z = πi.

(b) sinh z = cosh^{2} z. Exercise 4.11 Find the following:

(a) Log (−e^{2}(1 + i)).

(b) log (e^{−3}(1 − \sqrt{3}i)).

(c) Log(2 + i)^{2} − 2 Log(2 + i).

(d) Log(−2 + i)^{2} − 2 Log(−2 + i). Exercise 4.12 Explain why the set of values of log(i^{2}) is distinct from that of 2 log i. Exercise 4.13 What are all possible values of the expression Log(z_{1}z_{2}) − Log z_{1} − Log z_{2} as z_{1} and z_{2} range over the complex plane? Exercise 4.14 Find the analyticity domain of the following functions:

(a) Log((1 − 2i)z).

(b) \frac{Log((2 − i)z) + Log((2 + i)z)}{z^{3} − i} .

Exercise 4.15 Find all values of the following exponents, and indicate what the principal value is:

(a) (2i)^{2i}.

(b) (e^{2}(1 − \sqrt{3}i))^{5πi}.

(c) (3 + 2\sqrt{2}i)^{5/2}.

Exercise 4.16 Consider the function f(z) = z^{a} for a rational number a = p/q (in reduced terms).

(a) How many values does the function assume for any value of z?

(b) What is the image of the principal branch of the function?

5 Integrals

Exercise 5.1 Calculate the following integrals:

(a) ∫_{−1}^{1}(1 − (1 + i)t)^{3} dt .

(b) ∫_{2}^{3} (2i − \frac{1 + i}{t})^{2} dt .

(c) ∫_{0}^{π/4} e^{3it} dt .

(d) ∫_{0}^{π} cos at dt .

(e) ∫_{0}^{∞} e^{−at} dt , (Re a > 0).

Exercise 5.2 Prove the following orthogonality property: For integers m and n:

∫_{0}^{2π} e^{imθ} \overline{e^{inθ}} dθ = \{ \begin{matrix}0, & m ≠ n , \\ 2π, & m = n . \end{matrix}

Exercise 5.3 Verify the differentiation rule

(\frac{w_{1}(t)}{w_{2}(t)})′ = \frac{w_{2}(t)w′_{1}(t) − w_{1}(t)w′_{2}(t)}{w_{2}^{2}(t)}, (w_{2}(t) ≠ 0) .

Exercise 5.4 Let z = z(t), a ≤ t ≤ b, be a smooth arc. Let f(z) be defined along the arc and analytic at the point z(t_{0}) for some a ≤ t_{0} ≤ b. Show that, putting w(t) = f(z(t)), we have:

w′(t_{0}) = f′(z(t_{0}))z′(t_{0}) .

Exercise 5.5 Verify the validity of the inequality

|∫_{a}^{b} w(t) dt | ≤ ∫_{a}^{b} |w(t)| dt

for w(t) = (1 − 2i)e^{t}, a = 0, b = 2. Explain. Exercise 5.6 Consider the arc g(t), 0 ≤ t ≤ 1, given by

g(t) = \{ \begin{matrix}t + it sin \frac{1}{t}, & t > 0 , \\ 0, & t = 0 . \end{matrix}

(a) Prove that the arc is continuous.

(b) Prove that the arc is not smooth.

(c) What is the length of the arc?

(d) Will the arc be smooth if the it sin \frac{1}{t} term is replaced by it^{k} sin \frac{1}{t} with a sufficiently large integer k?

Exercise 5.7 Calculate ∫_{C} f(z) dz for the following functions and contours:

(a) f(z) = 3iz − 4 and C is the semicircle z(t) = 3e^{it}, 0 ≤ t ≤ π .

(b) f(z) = 3iz − 4 and C is the semicircle z(t) = 3e^{−it}, 0 ≤ t ≤ π .

(c) f(z) = 3iz − 4 and C is the broken line connecting the point 3 to the point −3 via the point 3i.

(d) f(z) = \frac{2z − 3}{z} and C is the circle of radius 3 around the point i, traversed counterclockwise.

(e) f(z) = e^{2\overline{z}} and C is the boundary of the square with vertices at the points ±1 ± i, traversed counter-clockwise.

(f) f(z) = \{ \begin{matrix}−2x, & x ≤ 0 , \\ x^{2} + ix, & x > , \end{matrix} and C is given by z(t) = t + it^{3}, −1 ≤ t ≤ 1.

Exercise 5.8 Show that for any positive numbers L and M, and for any complex number I with |I| ≤ LM, there exists a smooth arc C of length L and a continuous function f satisfying |f(z) ≤ M throughout C, such that ∫_{C} f(z) dz = I. Exercise 5.9 The arc C is given by:

r = θ, 0 ≤ θ ≤ 1 .

(a) Calculate the integral ∫_{C} |z| dz .

(b) Verify that the bound on the absolute value of the integral in terms of the length of C and a bound on the values of the integrand along the arc is satisfied.

Exercise 5.10 A circle of radius 1 is rotating along the positive real axis. Initially its center is located at the point i. The path traced by any point along the circumference of the circle is called a cycloid. Consider the cycloid generated by the point which is initially at 0, and let C be the arc of the cycloid from the point 0 to the point 2π.

(a) Show that C can be parametrized by

z(t) = t − sin t + i(1 − cos t), 0 ≤ t ≤ 2π .

(b) Calculate the arc length of C.

(c) Evaluate ∫_{C} \overline{z} dz .

Exercise 5.11 Prove the following inequalities without actually evaluating the integrals:

(a) |∫_{C}(z + \overline{z})e^{z} dz | ≤ 64e^{2}, where C is the triangle with vertices 2 − 3i, 2 + 3i and −2.

(b) |∫_{C} \frac{z dz}{z^{3} − 1}| ≤ \frac{4π}{7} , where C is the left half of the circle |z| = 2.

Exercise 5.12 Find upper bounds on |∫_{C} f(z) dz | for the following functions and contours:

(a) f(z) = \frac{1}{z^{2} + i} and C is the right half of the circle |z| = 4/3.

(b) f(z) = \frac{1}{z^{6}} and C is the line segment connecting the point 3i to the point 4.

Exercise 5.13 Let C be the circle of radius R around z_{0}, traversed counter-clockwise. Show that

∫_{C}(z − z_{0})^{a} dz = \{ \begin{matrix}2πi, & a = −1 , \\ 0, & a ∈ ℤ − \{−1\} , \\ \frac{2R^{a+1}}{a+1} sin((a + 1)π)i, & a ∈ ℝ − ℤ , \end{matrix}

where in the last case the principal branch is taken for both exponents.

Exercise 5.14 Show that:

(a) ∫_{C_{R}} \frac{z^{3} + 1}{z^{5} − 2} dz ⟶_{R→∞} 0, where C_{R} is the circle of radius R around 0.

(b) ∫_{S_{R}} \frac{(z^{5} − 3iz^{2} + 2) Log^{2} z}{(z^{4} − i)^{2}} dz ⟶_{R→∞} 0, where S_{R} is the square with vertices at the points

±R ± Ri.

Exercise 5.15 The integral ∫_{C} f(z) dz is evaluated using a parametrization z = z(t), a ≤ t ≤ b. Now suppose t = ϕ(τ), α ≤ τ ≤ β, where ϕ is differentiable and increasing in the interval [α, β], and satisfies ϕ(α) = a, ϕ(β) = b. Prove that, parametrizing C in terms of τ, we obtain the same value for the integral as with the original parametrization.

Exercise 5.16 Evaluate each of the following integrals using a primitive function, and check your result by a direct evaluation using a contour of your liking:

(a) ∫_{2}^{3i}(z^{2} − 2i) dz .

(b) ∫_{iπ}^{π}((2 + i) sin^{2} z − e^{z} cos z) dz .

Exercise 5.17 Employ the Cauchy-Goursat theorem to show that the integrals of the following functions along the circle of radius 2 around 0 all vanish:

(a) f(z) = \frac{1}{z − (3 − 4i)} .

(b) f(z) = cos ze^{iπz}.

(c) f(z) = \frac{iz − 1}{z^{2} − 8z + 25} .

(d) f(z) = z Log(z + 3).

Exercise 5.18 Employ Exercise 5.13 to evaluate ∫_{C}(z + 1 − 2i)^{n} dz , where C is the square with vertices 5, 5i, −5 and −5i, traversed counter-clockwise, for any integer n,

Exercise 5.19 Let C be the contour obtained as follows. It starts at 0 and goes all the way to 1. Then it changes its direction by 1 radian in the counter-clockwise direction and proceeds for a distance of 1/2. Next it again changes its direction by 1 radian in the counter-clockwise direction, and proceeds for a distance of 1/2^{2}, and so forth. Evaluate ∫_{C} Log(z + 2) dz . (Explain why, even though C is not smooth at infinitely many points, there is no ambiguity in this integral.)

Exercise 5.20 Evaluate the following integrals:

(a) ∫_{C} \frac{dz}{z} , where C is the arc from −3i to \frac{3(−1 + i)}{\sqrt{2}} on the circle |z| = 3.

(b) ∫_{C} z^{i} dz , where C is any contour from −1 to 1 lying (except for its end-points) in the

lower half-plane, and the principal branch of the exponent is taken in the integrand.

Exercise 5.21 Evaluate the integral ∫_{C} \frac{dz}{z} for the following contours C:

(a) C is the regular m-gon obained from the roots of unity ζ_{k} = e^{2πik/m}, k = 0, 1, …, m−1, of order m.

(b) C is the closed (but not simple) contour obtained by connecting ζ_{0} to ζ_{j}, ζ_{j} to ζ_{2j}, …, ζ_{(m−1)j} to ζ_{mj} (where the indices are to be understood modulo m). For which values of j is the integral undefined?

Exercise 5.22 Let f(z) denote the branch

f(z) = \sqrt[3]{r}e^{iθ/3}, z = re^{iθ}, r > 0, −π < θ < π,

of the function f(z) = \sqrt[3]{z}.

(a) Evaluate the integral ∫_{C} f(z) dz , where C is the closed contour connecting −2i to

2i along the right half of the circle |z| = 2 and returning from 2i to −2i along the imaginary axis.

(b) Explain why you cannot use Cauchy-Goursat theorem directly to prove that the integral in question vanishes.

(c) Show that, by changing the contour C by arbitrarily little you can obtain integrals to which Cauchy-Goursat theorem applies. Use this approach to conclude that the integral vanishes.

Exercise 5.23 Consider the integral

∫_{C} \frac{\overline{z} dz}{2i} ,

where C is any piecewise smooth simple closed contour.

(a) Show that, if C is the boundary of a rectangle with sides parallel to the coordinate axis, then the integral in question is the area of the rectangle enclosed by C.

(b) Explain why the same holds for any piecewise smooth C.

Exercise 5.24 Let f(z) be an entire function. Consider the integral

∫_{C} \frac{f(z) dz}{z^{5} − z} ,

where C may be any positively oriented simple closed contour along which the integrand is defined. How many distinct values can the integral take at most as C varies over all possible contours?

Exercise 5.25 Let f(z) be an entire function. Suppose it is known that

∫_{C_{1}} \frac{f(z) dz}{z(z − 1)} = 2 + 3i, ∫_{C_{2}} \frac{f(z) dz}{z(z − 1)} = 3 + 4i ,

where C_{1} and C_{2} are the positively oriented circles of radii 1/2 and 3/2, respectively, around 0. What are all possible values of

∫_{C} \frac{f(z) dz}{z(z − 1)}

as C ranges over all closed (simple or not, positively or negatively oriented) contours not passing through the points 0 and 1?

Exercise 5.26 Evaluate the following integrals by employing the Cauchy integral formula:

(a) ∫_{C} \frac{e^{2z}}{z} dz , where C is the positively oriented circle of radius 1 around 0.

(b) ∫_{C} \frac{sin z}{z^{3} − 1} dz , where C is the positively oriented circle of radius 1 around −1 + i.

(c) ∫_{C} \frac{Log(z + 5i)}{z^{4} + 1} dz , where C is the positively oriented circle of radius 2 around −1 + 2i.

(d) ∫_{C} \frac{sinh z}{z^{2} + π^{2}} dz , where C is the positively oriented circle of radius 5 around 3.

Exercise 5.27 Consider the integral ∫_{C} \frac{f(z)}{z^{m} − 1} dz , where f(z) is an entire function and m

a sufficiently large integer. Find all possible values of the integral if C is a positively oriented circle, not passing through any root of unity of order m, and whose radius is

(a) \frac{3}{m} .

(b) \frac{4}{m} .

(c) \frac{5}{m} .

Exercise 5.28 Carry out the induction step to complete the proof of the formula

f^{(n)}(z) = \frac{n!}{2πi} ∫_{C} \frac{f(s)}{(s − z)^{n+1}} ds .

Exercise 5.29 Let C be a simple closed contour, and let f(z) be a continuous function on C. A function g(z) is defined inside C by

g(z) = \frac{1}{2πi} ∫_{C} \frac{f(s)}{s − z} ds .

Prove that g(z) is analytic and satisfies

g′(z) = \frac{1}{2πi} ∫_{C} \frac{f(s)}{(s − z)^{2}} ds

at each point z inside C. Exercise 5.30 Evaluate the following integrals:

(a) ∫_{C} \frac{dz}{z^{6} + 2z^{3} + 1}, where C is the positively oriented circle of radius 2 around −2.

(b) ∫_{C} \frac{e^{(−1+i\sqrt{3})z}}{z^{16}} dz , where C is any positively oriented circle around 0.

(c) ∫_{C} \frac{(z^{2} − 1)^{5}}{z^{11}} dz , where C is any positively oriented circle around 0.

(d) ∫_{C} \frac{dz}{(z^{2} + 1)^{10}} , where C is the positively oriented circle of radius 1 around −i.

Exercise 5.31 Consider the integral ∫_{C} \frac{e^{(n+1)z} dz}{(z − 1)^{n+1}} , where C is the circle of radius 1 around the point 1.

(a) Evaluate the integral.

(b) Examine the asymptotics of your result as a function of n, and explain heuristically why this should be the asymptotics of the integral.

Exercise 5.32 Same as the preceding question for the integral ∫_{C} \frac{Log((n + 1)(1 + z)) dz}{(z − 1)^{n+1}} .

Exercise 5.33 Prove that a number z_{1} is a root of the polynomial P if and only if P (z) is divisible by z − z_{1}. Exercise 5.34 Let P be a polynomial with real coefficients.

(a) Prove that, if a complex number z_{1} is a root of P, then so is \overline{z_{1}}. Moreover, z_{1} and \overline{z_{1}} are roots of equal multiplicity.

(b) Use the preceding part to prove that, if P has an odd degree, then it has at least one real root.

(c) Prove the same result without using the first part of the question. Exercise 5.35 Let P be a non-constant polynomial in two variables.

(a) Prove that the equation P (z, w) = 0 has infinitely many solutions.

(b) Does the equation P (z, \overline{z}) = 0 necessarily have solutions?

6 Series

Exercise 6.1 Consider the set of all infinite sequences (z_{n})_{n=1}^{∞}.

(a) Show that, with componentwise addition and multiplication by scalar, it forms a linear space.

(b) Show that its subset, consisting of all sequences for which the series ∑_{n=1}^{∞} z_{n} is convergent, is a linear subspace.

(c) Is this subspace finite-dimensional or infinite-dimensional?

(d) Show that the mapping, taking each sequence (z_{n}) in the subspace to the sum of the corresponding series, forms a linear functional.

Exercise 6.2 Suppose ∑_{n=1}^{∞} z_{n} = S. Prove that ∑_{n=1}^{∞} \overline{z_{n}} = \overline{S}. Exercise 6.3 A series ∑_{n=1}^{∞} z_{n} is absolutely convergent if the series ∑_{n=1}^{∞} |z_{n}| is convergent. Prove that an absolutely convergent series is convergent. Exercise 6.4 Consider the following possible property of functions h : ℂ ⟶ ℂ: For any sequence (z_{n}) such that the series ∑_{n=1}^{∞} z_{n} converges, the series ∑_{n=1}^{∞} h(z_{n}) converges as well.

(a) Prove that, if h has the property in question, then it vanishes at the point 0 and is continuous there.

(b) Prove that the set of all functions h having this property is a linear space with respect to pointwise addition and multiplication by scalar.

(c) Which function h possesses this property according to Exercise 6.2?

(d) Does the function h(z) = z^{2} satisfy this property?

(e) How about the function h(z) = z^{3}? Exercise 6.5 Prove that, if f and g are analytic at a point z, then

(f(z)g(z))^{(n)} = ∑_{k=0}^{n} (\begin{matrix}n \\ k \end{matrix}) f^{(k)}(z)g^{(n−k)}(z), n = 0, 1, 2, ….

Exercise 6.6 Find the Maclaurin expansion of the following functions directly (i.e., without using the expansion of well-known functions). In each case, note the region in which the expansion is valid.

(a) f(z) = z^{2} cosh z.

(b) f(z) = \frac{z^{3}}{1 − z}.

(c) f(z) = z^{2} Log(1 + z).

(d) f(z) = g(1 + z), where g(z) = \sqrt[4]{r}e^{iθ/4} for r ≥ 0 and −π < θ < π. Exercise 6.7 Find the Maclaurin expansion of the following functions starting with that of well-known functions. Note the region of convergence in each case.

(a) f(z) = z^{3} sin z.

(b) f(z) = z (e^{z^{2}} + 1)^{3}.

(c) f(z) = \frac{Log(1 + z)}{1 − 2z}.

Exercise 6.8 Find the Taylor expansion of the following functions about the indicated points. Note the region of convergence.

(a) f(z) = sin z, z_{0} = π/4.

(b) f(z) = sinh z, z_{0} = πi.

(c) f(z) = Log z, z_{0} = 2i.

(d) f(z) = e^{2z}, z_{0} = 1 + πi/3.

(e) f(z) = \sqrt[3]{r}e^{iθ/3} for r ≥ 0 and −π < θ < π, z_{0} = 3 − 4i. Exercise 6.9 Find all Laurent expansion for the following functions about the indicated points. Note the region of convergence of each.

(a) f(z) = \frac{z}{z − 2i}, z_{0} = 2i.

(b) f(z) = \frac{1}{(z − 2)(z − 3)}, z_{0} = 3.

(c) f(z) = \frac{1}{z^{3} − z} , z_{0} = 0.

(d) f(z) = \frac{1}{z^{2} − 6z + 25}, z_{0} = 0.

(e) f(z) = \frac{1}{z^{5} + z} , z_{0} = 0.

(f) f(z) = \frac{(z^{2} + 1)e^{1/z}}{z} , z_{0} = 0.

(g) f(z) = \frac{sin \frac{1}{z}}{z^{3} + z} , z_{0} = 0.

(h) f(z) = (z^{3} − z^{2}) sin \frac{1}{z − 2}, z_{0} = 2.

Exercise 6.10 For each of the expansions you found in the preceding question, write down an integral whose value can be easily deduced from the Laurent expansion of that function.