Cauchy's integral theorem

Theorem 4..2 (Cauchy's integral theorem)   When the function $f(z)$ is analytic in the region $\Omega$, let $C$ be any single closed curve in $\Omega$, and the region $\Omega_{1}$ surrounded by $C$ is inside $\Omega$. Then

$$\int_{C}f(z)\;dz = 0$$

Exercise4.3

1. Prove the followings:

(a)

The function $f(z)$ is analytic in the region $D$, and the two curves $C_{1}, C_{2}$ connecting the two points $a$ and $b$ are in $D$ and if the area enclosed by $C_{1}, C_{2}$ is in $D$, then

$$\int_{c_{1}}f(z)dz = \int_{c_{2}}f(z)dz$$

(b)

If $f(z)$ is analytic in the region $D$ surrounded by two single closed curves $C_{1}, C_{2}$, then

$$\int_{C_{1}}f(z)dz = \int_{C_{2}}f(z) dz$$

2. Integrate the following function along the shown closed curve.

(a)

$\frac{1}{z^2 + 1}, C:$ centered at the origin and a circle with the radius $r > 1$.

(b)

$\frac{z}{(2z + i)(z - 2)}, C:$ unit circle

(c)

$\frac{1}{z^4}, C:$ centered at the origin and a semicircle with the radius $r > 1$ and the diameter on the real axis.

3. Find the following integral. The integration path is a line segment connecting the lower end and the upper end.

(a)

$\int_{i}^{1}z^2 dz$

(b)

$\int_{0}^{i}ze^{z} dz$

(c)

$\int_{0}^{1+i}\frac{z}{z+1} dz$

(d)

$\int_{0}^{i}\frac{1}{\sqrt{1 - z^2}} dz$

4. Prove the following functions are harmonic functions and Create a holomorphic function that has it in the real part.

(a)

$u = x^2 - y^2$

(b)

$u = e^{x}\cos{y}$

(c)

$u = \cos{x}\sinh{y}$

(d)

$u = \frac{1}{2}\log_{e}(x^2 + y^2)$