• Home
  • Textbooks
  • Complex Analysis
  • Integral Calculus in the Complex Plane $\mathbb{C}$

Complex Analysis

Eberhard Freitag, Rolf Busam

Chapter 2

Integral Calculus in the Complex Plane $\mathbb{C}$ - all with Video Answers

Educators

Section 1

Complex Line Integrals

00:51

Problem 1

The figure on the right shows a closed curve $\alpha$, Give an explicit parametrization for $\alpha$ and calculate
$$
\frac{1}{2 \pi \mathrm{i}} \int_{\alpha} \frac{1}{z} d z
$$

Arun Bana
Arun Bana
Numerade Educator
06:17

Problem 2

Let $\alpha:[0, \pi] \rightarrow \mathbb{C}$ be defined by
$$
\alpha(t):=\exp (\mathrm{i} t)
$$
and $\beta:[0,2] \rightarrow \mathbb{C}$ by
$$
\beta(t)= \begin{cases}1+t(-\mathrm{i}-1) & \text { for } t \in[0,1] \\ 1-t+\mathrm{i}(t-2) & \text { for } t \in[1,2]\end{cases}
$$
Sketch $\alpha$ and $\beta$, and calculate
$$
\int_{\alpha} \frac{1}{z} d z \quad \text { and } \quad \int_{\beta} \frac{1}{z} d z
$$

Victor Salazar
Victor Salazar
Numerade Educator
02:11

Problem 3

Prove the transformation invariance of the line integral, II.1.5, (4).

Vikash Ranjan
Vikash Ranjan
Numerade Educator
02:03

Problem 4

Sketch the following curve $\alpha$ ("figure eight")
$$
\alpha(t):=\left\{\begin{aligned}
1-\exp (\text { it }) & \text { for } t \in[0,2 \pi] \\
-1+\exp (-\mathrm{i} t) & \text { for } t \in[2 \pi, 4 \pi]
\end{aligned}\right.
$$

Km Neeraj
Km Neeraj
Numerade Educator
01:47

Problem 5

Compute
$$
\int_{\alpha} z \exp \left(z^{2}\right) d z
$$
where
(a) $\alpha$ is the line between the point 0 and the point $1+i$,
(b) $\alpha$ is the piece of the parabola with equation $y=x^{2}$, which lies between the points 0 and $1+\mathrm{i}$

Narayan Hari
Narayan Hari
Numerade Educator
01:39

Problem 6

Compute
$$
\int_{\alpha} \sin z d z
$$
where $\alpha$ is the piece of the parabola with equation $y=x^{2}$, which lies between the points 0 and $-1+i$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
06:19

Problem 7

Let $[a, b]$ and $[c, d]$ ( $a<b$ and $c<d$ ) be compact intervals in $\mathbb{R}$. Show: There is an affine map
$$
\begin{aligned}
\varphi:[a, b] & \longrightarrow[c, d] \\
t & \longmapsto \alpha t+\beta
\end{aligned}
$$
with $\varphi(a)=c$ and $\varphi(b)=d$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:37

Problem 8

Let $R>0$ be a positive number. We consider the curve
$$
\beta(t)=R \exp (i t), \quad 0 \leq t \leq \frac{\pi}{4}
$$
Show:
$$
\left|\int_{\beta} \exp \left(\mathrm{i} z^{2}\right) d z\right| \leq \frac{\pi\left(1-\exp \left(-R^{2}\right)\right)}{4 R}<\frac{\pi}{4 R}
$$

Lucas Finney
Lucas Finney
Numerade Educator
01:46

Problem 9

Let $\alpha:[a, b] \rightarrow \mathbb{C}$ be continuously differentiable and assume that the function $f:$ Image $\alpha \rightarrow \mathbb{C}$ is continuous.
Show: For any $\varepsilon>0$ there exists a $\delta>0$ with the following property: If $\left\{a_{0}, \ldots, a_{N}\right\}$ and $\left\{c_{1}, \ldots, c_{N}\right\}$ are finite subsets of $[a, b]$ with
$$
a=a_{0} \leq c_{1} \leq a_{1} \leq c_{2} \leq a_{2} \leq \cdots \leq a_{N-1} \leq c_{N} \leq a_{N}=b
$$
and
$$
a_{\nu}-a_{\nu-1}<\delta \text { for } \nu=1, \ldots, N
$$
then
$$
\left|\int_{\alpha} f(z) d z-\sum_{\nu=1}^{N} f\left(\alpha\left(c_{\nu}\right)\right) \cdot\left(\alpha\left(a_{\nu}\right)-\alpha\left(a_{\nu-1}\right)\right)\right|<\epsilon
$$
(Approximation of the line integral by a RIEMANN sum.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:46

Problem 10

Let $\alpha:[a, b] \rightarrow \mathbb{C}$ be continuously differentiable and assume that the function $f:$ Image $\alpha \rightarrow \mathbb{C}$ is continuous.
Show: For any $\varepsilon>0$ there exists a $\delta>0$ with the following property: If $\left\{a_{0}, \ldots, a_{N}\right\}$ and $\left\{c_{1}, \ldots, c_{N}\right\}$ are finite subsets of $[a, b]$ with
$$
a=a_{0} \leq c_{1} \leq a_{1} \leq c_{2} \leq a_{2} \leq \cdots \leq a_{N-1} \leq c_{N} \leq a_{N}=b
$$
and
$$
a_{\nu}-a_{\nu-1}<\delta \text { for } \nu=1, \ldots, N
$$
then
$$
\left|\int_{\alpha} f(z) d z-\sum_{\nu=1}^{N} f\left(\alpha\left(c_{\nu}\right)\right) \cdot\left(\alpha\left(a_{\nu}\right)-\alpha\left(a_{\nu-1}\right)\right)\right|<\epsilon
$$
(Approximation of the line integral by a RIEMANN sum.)

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:15

Problem 11

A smooth curve is called regular if its derivative does not vanish anywhere. Assume that there are given an analytic function $f: D \rightarrow \mathbb{C}, D \subset \mathbb{C}$ open, and a point $a \in D$ with $f^{\prime}(a) \neq 0$, and also two regular curves $\alpha, \beta:[-1,1] \rightarrow D$ with $\alpha(0)=\beta(0)=a$. One may then consider the oriented angle $\angle\left(\alpha^{\prime}(0), \beta^{\prime}(0)\right)$ (see I.1, Exercise 4). This is the angle between the two intersecting curves. Show that the two image curves $f \circ \alpha$ and $f \circ \beta$ intersect with the same angle at their intersection point $f(a)=f(\alpha(0))=f(\beta(0))$.

Brittany Knowlton
Brittany Knowlton
Numerade Educator