Let's go through each part of the problem step by step:

(b)(i) Establishing the Cauchy-Riemann equations for the complex function $w = u(x, y) + i v(x, y)$ in terms of $r$ and $\theta$

We are given:

• $x(r, \theta) = r \cos \theta$
• $y(r, \theta) = r \sin \theta$

To establish the Cauchy-Riemann equations, we need to express $u(x, y)$ and $v(x, y)$ in terms of $r$ and $\theta$. Let's assume $w = u(x, y) + i v(x, y)$ is a complex function, and we're looking for the conditions that make $w$ analytic.

Recall that the Cauchy-Riemann equations in Cartesian coordinates are:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Now, we need to rewrite the partial derivatives with respect to $x$ and $y$ in terms of $r$ and $\theta$.

Since $x = r \cos \theta$ and $y = r \sin \theta$, we can compute the following derivatives:

1. Partial derivatives of $x$ and $y$ with respect to $r$ and $\theta$:

$\frac{\partial x}{\partial r} = \cos \theta, \quad \frac{\partial x}{\partial \theta} = -r \sin \theta$ $\frac{\partial y}{\partial r} = \sin \theta, \quad \frac{\partial y}{\partial \theta} = r \cos \theta$

Using the chain rule, the Cauchy-Riemann equations become:

$\frac{\partial u}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial u}{\partial \theta} \frac{\partial \theta}{\partial x} = \frac{\partial v}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial v}{\partial \theta}$