Given points $O(0)$ and $A(1+i)$ in the complex plane, find the complex number $w$ which is the reflection of the complex number $z$ across the line $OA$. Express $w$ in terms of $z$.

1. Problem Description

Given points $O(0)$ and $A(1+i)$ in the complex plane, find the complex number $w$ which is the reflection of the complex number $z$ across the line $OA$. Express $w$ in terms of $z$.

2. Solution Steps

The line $OA$ has the argument $\arg(1+i) = \frac{\pi}{4}$.

To reflect a complex number $z$ across the line $OA$, we can perform the following steps:

1. Rotate $z$ by $-\frac{\pi}{4}$ so that the line $OA$ coincides with the real axis. This gives us $z e^{-i\pi/4}$.

2. Take the complex conjugate of the result. This gives us $\overline{z e^{-i\pi/4}} = \bar{z}e^{i\pi/4}$.

3. Rotate back by $\frac{\pi}{4}$. This gives us $\bar{z} e^{i\pi/4} e^{i\pi/4} = \bar{z} e^{i\pi/2} = \bar{z} i$.

Therefore, $w = i\bar{z}$.

The rotation can also be written as:

Let $\theta = \frac{\pi}{4}$

$w = e^{i\theta} \overline{e^{-i\theta} z} = e^{i\theta} e^{i\theta} \bar{z} = e^{2i\theta} \bar{z} = e^{i \frac{\pi}{2}} \bar{z} = i\bar{z}$

3. Final Answer

$w = i\bar{z}$