Question

(a) Let w = f(z) = z^2. Determine the images of the curves $\gamma_1$ and $\gamma_2$ under f, where $\gamma_1 = \{x + iy \in \mathbb{C}: x \ge 0, y = 0\} \cup \{x + iy: x = 0, y \ge 0\}$, $\gamma_2 = \{x + iy \in \mathbb{C}: x > 0, y > 0\} \cap \{x + iy: xy = 1\}$. (b) Let U be the domain between $\gamma_1$ and $\gamma_2$. Show that the set f(U) = \{f(z): z \in U\} is a strip in the w-plane. (c) Hence, or otherwise, solve the following Dirichlet problem: $\begin{cases} \nabla^2 \phi(x, y) = 0, & \text{for all } x + iy \in U,\\ \phi(x, y) = 0, & \text{for all } x + iy \text{ in } \gamma_1,\\ \phi(x, y) = 1, & \text{for all } x + iy \text{ in } \gamma_2. \end{cases}$ [Hint: It is easier to solve the Dirichlet problem in the strip than the half plane when each boundary of the strip is kept constant.]

          (a) Let
w = f(z) = z^2.
Determine the images of the curves \(\gamma_1\) and \(\gamma_2\) under f, where
\(\gamma_1 = \{x + iy \in \mathbb{C}: x \ge 0, y = 0\} \cup \{x + iy: x = 0, y \ge 0\}\),
\(\gamma_2 = \{x + iy \in \mathbb{C}: x > 0, y > 0\} \cap \{x + iy: xy = 1\}\).
(b) Let U be the domain between \(\gamma_1\) and \(\gamma_2\). Show that the set
f(U) = \{f(z): z \in U\}
is a strip in the w-plane.
(c) Hence, or otherwise, solve the following Dirichlet problem:
\(\begin{cases}
\nabla^2 \phi(x, y) = 0, & \text{for all } x + iy \in U,\\
\phi(x, y) = 0, & \text{for all } x + iy \text{ in } \gamma_1,\\
\phi(x, y) = 1, & \text{for all } x + iy \text{ in } \gamma_2.
\end{cases}\)
[Hint: It is easier to solve the Dirichlet problem in the strip than the
half plane when each boundary of the strip is kept constant.]

$(a) Let w = f(z) = z^2. Determine the images of the curves γ1 and γ2 under f, where γ1 = {x + iy ∈ℂ: x ≥ 0, y = 0}∪{x + iy: x = 0, y ≥ 0}, γ2 = {x + iy ∈ℂ: x > 0, y > 0}∩{x + iy: xy = 1}. (b) Let U be the domain between γ1 and γ2. Show that the set f(U) = {f(z): z ∈U} is a strip in the w-plane. (c) Hence, or otherwise, solve the following Dirichlet problem: ∇^2 ϕ(x, y) = 0, for all x + iy ∈ U, ϕ(x, y) = 0, for all x + iy in γ1, ϕ(x, y) = 1, for all x + iy in γ2. [Hint: It is easier to solve the Dirichlet problem in the strip than the half plane when each boundary of the strip is kept constant.]$

Added by Albert W.

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