Exercises

Compute \(\int_\gg \frac{ \diff{z} }{ z }\) where \(\gg\) is the unit circle, oriented counterclockwise. More generally, show that for any \(w \in \C\) and \(r > 0\text{,}\)

\begin{equation*} \int_{ C[w,r] } \frac{\diff{z}}{z-w} \ = \ 2 \pi i \,\text{.} \end{equation*}

🔗

5.

Integrate the following functions over the circle \(C[0,2]\text{:}\)

🔗

(a)

\(f(z) = z + \overline z\)

🔗

Answer.

\(8 \pi i\)

🔗

(b)

\(f(z) = z^2 - 2z + 3\)

🔗

Answer.

\(0\)

🔗

(c)

\(f(z) = \frac 1 { z^4 }\)

🔗

Answer.

\(0\)

🔗

(d)

\(f(z) = xy\)

🔗

Answer.

\(0\)

🔗

6.

Evaluate the integrals \(\int_\gamma x\,\diff{z}\text{,}\) \(\int_\gamma y\,\diff{z}\text{,}\) \(\int_\gamma z\,\diff{z}\) and \(\int_\gamma \conj z\,\diff{z}\) along each of the following paths.

🔗

(a)

\(\gamma\) is the line segment from \(0\) to \(1-i\)

🔗

Answer.

\(\frac 1 2 (1-i)\text{,}\) \(\frac 1 2 (i-1)\text{,}\) \(-i\text{,}\) \(1\)

🔗

(b)

\(\gamma = C[0,1]\)

🔗

Answer.

\(\pi i\text{,}\) \(-\pi\text{,}\) 0, \(2 \pi i\)

🔗

(c)

\(\gamma = C[a,r]\) for some \(a \in \C\)

🔗

Answer.

\(\pi i r^2\text{,}\) \(-\pi r^2\text{,}\) 0, \(2 \pi i r^2\)

🔗

7.

Evaluate \(\int_\gamma \exp(3z)\,\diff{z}\) for each of the following paths:

🔗

(a)

\(\gamma\) is the line segment from \(1\) to \(i\)

🔗

Answer.

\(\frac 1 3 (e^{ 3i }-e^3)\)

🔗

(b)

\(\gamma = C[0,3]\)

🔗

Answer.

\(0\)

🔗

(c)

\(\gamma\) is the arc of the parabola \(y=x^2\) from \(x=0\) to \(x=1\)

🔗

Answer.

\(\frac 1 3 (\exp(3+3i) - 1)\)

🔗

8.

Compute \(\int_\gg f\) for the following functions \(f\) and paths \(\gg\text{:}\)

🔗

(a)

\(f(z) = z^2\) and \(\gamma(t)=t+it^2\text{,}\) \(0\le t\le1\text{.}\)

🔗

(b)

\(f(z) = z\) and \(\gamma\) is the semicircle from \(1\) through \(i\) to \(-1\text{.}\)

🔗

(c)

\(f(z) = \exp(z)\) and \(\gamma\) is the line segment from \(0\) to a point \(z_0\text{.}\)

🔗

(d)

\(f(z) = |z|^2\) and \(\gamma\) is the line segment from \(2\) to \(3+i\text{.}\)

🔗

(e)

\(f(z) = z + \frac 1 z\) and \(\gamma\) is parametrized by \(\gamma(t)\text{,}\) \(0\leq t\leq 1\text{,}\) and satisfies \(\Im \gamma(t) > 0\text{,}\) \(\gamma(0) = -4+i\text{,}\) and \(\gamma(1) = 6+2i\text{.}\)

🔗

(f)

\(f(z) = \sin(z)\) and \(\gamma\) is some piecewise smooth path from \(i\) to \(\pi\text{.}\)

🔗

9.

Prove Proposition 4.1.3 and the fact that the length of \(\gg\) does not change under reparametrization.

🔗

Hint.

Assume \(\gg\text{,}\) \(\sigma\text{,}\) and \(\tau\) are smooth. Start with the definition of \(\int_\sigma f\text{,}\) apply the chain rule to \(\sigma=\gg\circ\tau\text{,}\) and then use the change of variables formula, Theorem A.0.6.

🔗

10.

Prove the following integration by parts statement: Let \(f\) and \(g\) be holomorphic in \(G\text{,}\) and suppose \(\gg \subset G\) is a piecewise smooth path from \(\gg(a)\) to \(\gg(b)\text{.}\) Then

\begin{equation*} \int_\gg f g' \ = \ f(\gg(b)) g(\gg(b)) - f(\gg(a)) g(\gg(a)) - \int_\gg f' g \, \text{.} \end{equation*}

🔗

11.

Let \(\ds I(k) := \frac{1}{2\pi}\int_0^{2\pi} e^{ikt}\, \diff{t}\text{.}\)

🔗

(a)

Show that \(I(0) = 1\text{.}\)

🔗

(b)

Show that \(I(k) = 0\) if \(k\) is a nonzero integer.

🔗

(c)

What is \(I(\frac 1 2)\text{?}\)

🔗

12.

Compute \(\ds \int_{ C[0,2] } z^{\frac 1 2 } \,\diff{z} \text{.}\)

🔗

Answer.

\(- \frac{ 8 \sqrt 2 }{ 3 }\)

🔗

13.

Show that \(\int_\gg z^n \, \diff{z}=0\) for any closed piecewise smooth \(\gg\) and any integer \(n\ne-1\text{.}\) (If \(n\) is negative, assume that \(\gg\) does not pass through the origin, since otherwise the integral is not defined.)

🔗

14.

Exercise 4.5.13 excluded \(n=-1\) for a good reason: Exercise 4.5.4 gives a counterexample. Generalizing these, if \(m\) is any integer, find a closed path \(\gg\) so that \(\int_\gg z^{-1}\,\diff{z}=2m\pi i\text{.}\)

🔗

15.

Taking the previous two exercises one step further, fix \(z_0 \in \C\) and let \(\gg\) be a simple, closed, positively oriented, piecewise smooth path such that \(z_0\) is inside \(\gg\text{.}\) Show that, for any integer \(n\text{,}\)

\begin{equation*} \int_\gg (z-z_0)^n \, \diff{z} \ = \ \begin{cases}2 \pi i \amp \text{ if } n = -1 \, , \\ 0 \amp \text{ otherwise. } \end{cases} \end{equation*}

🔗

16.

Prove that \(\int_\gg z \exp(z^2) \, \diff{z} = 0\) for any closed path \(\gg\text{.}\)

🔗

17.

Show that \(F(z)=\frac i2\Log(z+i) -\frac i2\Log(z-i)\) is an antiderivative of \(\frac1{1+z^2}\) for \(\Re(z)>0\text{.}\) Is \(F(z)\) equal to \(\arctan z\text{?}\)

🔗

18.

Compute the following integrals, where \(\gg\) is the line segment from 4 to \(4i\text{.}\)

🔗

(a)

\(\ds \int_\gg \frac{ z+1 }{ z } \, \diff{z}\)

🔗

Answer.

\(-4 + i (4 + \frac \pi 2)\)

🔗

(b)

\(\ds \int_\gg \frac{ \diff{z} }{ z^2 + z }\)

🔗

Answer.

\(\ln(5) - \frac 1 2 \ln(17) + i ( \frac \pi 2 - \Arg(4i+1) )\)

🔗

(c)

\(\ds \int_\gg z^{ - \frac 1 2 } \, \diff{z}\)

🔗

Answer.

\(2 \sqrt 2 - 4 + 2 \sqrt 2 \, i\)

🔗

(d)

\(\ds \int_\gg \sin^2(z) \, \diff{z}\)

🔗

Answer.

\(\frac 1 4 \sin(8) - 2 + i \left( 2 - \frac 1 4 \sinh(8) \right)\)

🔗

19.

Compute the following integrals.

🔗

(a)

\(\displaystyle \int_{\gg_1} z^i \, \diff{z}\) where \(\gg_1 (t) = e^{it} , \ - \frac{\pi}{2} \leq t \leq \frac{\pi }{2}\text{.}\)

🔗

(b)

\(\displaystyle \int_{\gg_2} z^i \, \diff{z}\) where \(\gg_2 (t) = e^{it} , \ \frac{\pi}{2} \leq t \leq \frac{3\pi} {2}\text{.}\)

🔗

20.

Show that (4.4) gives a homotopy between the unit circle and the square with vertices \(\pm 3 \pm 3i\text{.}\)

🔗

21.

Use Exercise 1.5.34 give a homotopy that is an alternative to (4.4) and does not need a piecewise definition.

🔗

22.

Suppose \(a \in \C\) and \(\gamma_0\) and \(\gamma_1\) are two counterclockwise circles so that \(a\) is inside both of them. Give a homotopy that proves \(\gg_0 \sim_{ \C \setminus \{ a \} } \gg_1\text{.}\)

🔗

23.

Prove that \(\sim_G\) is an equivalence relation.

🔗

24.

Suppose that \(\gg\) is a closed path in a region \(G\text{,}\) parametrized by \(\gg(t),\,t\in[0,1]\text{,}\) and \(\tau\) is a continuous increasing function from \([0,1]\) onto \([0,1]\text{.}\) Show that \(\gg\) is \(G\)-homotopic to the reparametrized path \(\gg\circ\tau\text{.}\)

🔗

Hint.

Make use of \(\tau_s(t)=s\tau(t)+(1-s)t\) for \(0\le s\le1\text{.}\)

🔗

25.

(a)

Prove that any closed path is \(\C\)-contractible.

🔗

(b)

Prove that any two closed paths are \(\C\)-homotopic.

🔗

26.

This exercise gives an alternative proof of Corollary 4.3.8 via Green’s Theorem A.0.10. Suppose \(G \subseteq \C\) is a region, \(f\) is holomorphic in \(G\text{,}\) \(f'\) is continuous, \(\gg\) is a simple piecewise smooth closed curve, and \(\gg\sim_G0\text{.}\) Explain that we may write

\begin{equation*} \int_\gg f(z) \, \diff{z} \ = \ \int_\gg (u + i \, v) (\diff{x} + i \, \diff{y}) \ = \ \int_\gg u \, \diff{x} - v \, \diff{y} \ + \ i \int_\gg v \, \diff{x} + u \, \diff{y} \end{equation*}

and show that these integrals vanish, by using Green’s Theorem A.0.10 together with Proposition 4.4.6, and then the Cauchy–Riemann equations (2.2).

🔗

27.

Fix \(a \in \C\text{.}\) Compute

\begin{equation*} I(r) \ := \ \int_{ C[0,r] } \frac{ \diff{z} }{ z-a } \,\text{.} \end{equation*}

(You should get different answers for \(r\lt \abs a\) and \(r>\abs a\text{.}\))

🔗

Hint.

In one case \(\gamma_r\) is contractible in \(\C\setminus\listset a\text{.}\) In the other you can combine Exercise 4.5.4 and Exercise 4.5.22.

🔗

Answer.

\(0\) for \(r\lt \abs a\text{;}\) \(2\pi i\) for \(r \gt \abs a\)

🔗

28.

Suppose \(p(z)\) is a polynomial in \(z\) and \(\gg\) is a closed piecewise smooth path in \(\C\text{.}\) Show that \(\int_\gg p \ = \ 0 \, \text{.}\)

🔗

29.

Show that

\begin{equation*} \int_{ C[0,2] } \frac{ \diff{z} }{ z^3 + 1 } \ = \ 0 \end{equation*}

by arguing that this integral does not change if we replace \(C[0,2]\) by \(C[0,r]\) for any \(r > 1\text{,}\) then use Proposition 4.1.8 Item d to obtain an upper bound for \(|\int_{ C[0,r] } \frac{ \diff{z} }{ z^3 + 1 }|\) that goes to 0 as \(r \to \infty\text{.}\)

🔗

30.

Compute the real integral

\begin{equation*} \int_0^{2 \pi} \frac{\diff\phi}{2 + \sin \phi} \end{equation*}

by writing the sine function in terms of the exponential function and making the substitution \(z = e^{i \phi}\) to turn the real integral into a complex integral.

🔗

Answer.

\(\frac{2 \pi }{ \sqrt 3 }\)

🔗

31.

Prove that for \(0 \lt r \lt 1\text{,}\)

\begin{equation*} \frac 1 {2 \pi} \int_0^{ 2 \pi } \frac{ 1-r^2 }{ 1 - 2r \cos(\phi) + r^2 } \, \diff\phi \ = \ 1 \, \text{.} \end{equation*}

(The function \(P_r(\phi) := \frac{ 1-r^2 }{ 1 - 2r \cos(\phi) + r^2 }\) is the Poisson kernel

Named after Siméon Denis Poisson (1781–1840).

and plays an important role in the world of harmonic functions, as we will see in Exercise 6.3.13.)

🔗

32.

Suppose \(f\) and \(g\) are holomorphic in the region \(G\) and \(\gg\) is a simple piecewise smooth \(G\)-contractible path. Prove that if \(f(z) = g(z)\) for all \(z \in \gg\text{,}\) then \(f(z) = g(z)\) for all \(z\) inside \(\gg\text{.}\)

🔗

33.

Show that Corollary 4.3.8, for simple paths, is also a corollary of Theorem 4.4.5.

🔗

34.

Compute

\begin{equation*} I(r) \ := \ \int_{ C[-2i,r] } \frac{ \diff{z} }{ z^2 + 1 } \end{equation*}

for \(r \ne 1, \, 3\text{.}\)

🔗

Answer.

\(0\)

🔗

35.

Find

\begin{equation*} \int_{ C[0,r] }\frac{\diff{z}}{z^2-2z-8} \end{equation*}

for \(r=1\text{,}\) \(r=3\) and \(r=5\text{.}\)

🔗

Hint.

Compute a partial-fractions expansion of the integrand.

🔗

Answer.

\(0\) for \(r=1\text{;}\) \(- \frac{\pi i} 3\) for \(r = 3\text{;}\) \(0\) for \(r=5\)

🔗

36.

Use the Cauchy Integral Formula (Theorem 4.4.1) to evaluate the integral in Exercise 4.5.35 when \(r=3\text{.}\)

🔗

37.

Compute the following integrals.

🔗

(a)

\(\ds \int_{ C[-1,2] } \frac{z^2}{4-z^2} \, \diff{z}\)

🔗

Answer.

\(2 \pi i\)

🔗

(b)

\(\ds \int_{ C[0,1] }\frac{\sin z}{z} \, \diff{z}\)

🔗

Answer.

\(0\)

🔗

(c)

\(\ds \int_{ C[0,2] } \frac{\exp(z)}{z(z-3)} \, \diff{z}\)

🔗

Answer.

\(- \frac{ 2 \pi i }{ 3 }\)

🔗

(d)

\(\ds \int_{ C[0,4] } \frac{\exp(z)}{z(z-3)} \, \diff{z}\)

🔗

Answer.

\(\frac{ 2 \pi i }{ 3 } ( e^3 - 1 )\)

🔗

38.

Let \(f(z) = \frac{ 1 }{ z^2 - 1 }\) and define the two paths \(\gg = C[1,1]\) oriented counter-clockwise and \(\sigma = C[-1,1]\) oriented clockwise. Show that \(\int_\gg f \ = \ \int_\sigma f\) even though \(\gg \not\sim_G \sigma\) where \(G = \C \setminus \{ \pm 1 \}\text{,}\) the region of holomorphicity of \(f\text{.}\)

🔗

39.

This exercise gives an alternative proof of Cauchy’s Integral Formula (Theorem 4.4.5) that does not depend on Cauchy’s Theorem (Theorem 4.3.4). Suppose the region \(G\) is convex; this means that, whenever \(z\) and \(w\) are in \(G\text{,}\) the line segment between them is also in \(G\text{.}\) Suppose \(f\) is holomorphic in \(G\text{,}\) \(f'\) is continuous, and \(\gg\) is a positively oriented, simple, closed, piecewise smooth path, such that \(w\) is inside \(\gg\) and \(\gg \sim_G 0\text{.}\)

🔗

(a)

Consider the function \(g: [0,1] \to \C\) given by

\begin{equation*} g(t) \ := \ \int_\gg \frac{ f \left( w + t(z-w) \right) }{ z-w } \, \diff{z} \, \text{.} \end{equation*}

Show that \(g'=0\text{.}\)

🔗

Hint.

Use Theorem A.0.9 (Leibniz’s rule) and then find an antiderivative for \(\frac{ \partial f }{ \partial t } \left( w + t(z-w) \right)\text{.}\)

🔗

(b)

Prove Theorem 4.4.5 by evaluating \(g(0)\) and \(g(1)\text{.}\)

🔗

(c)

Why did we assume \(G\) is convex?

🔗

Prev Top Next