07/17/2022, 06:46 AM
Yes, Mphlee
Also remember I chose \(0 < \lambda < 1\) in this example. If \(0 < | \lambda| < 1\), then the half plane \(H\) is rotated such that \(|\lambda^z| < 1\). So it no longer becomes the real part dominates it, you have to do a slight coordinate change first. Exact same principles across the board though. You can still bound based on the real argument, you just have to be more careful, because the imaginary argument can grow unbounded.
Also remember I chose \(0 < \lambda < 1\) in this example. If \(0 < | \lambda| < 1\), then the half plane \(H\) is rotated such that \(|\lambda^z| < 1\). So it no longer becomes the real part dominates it, you have to do a slight coordinate change first. Exact same principles across the board though. You can still bound based on the real argument, you just have to be more careful, because the imaginary argument can grow unbounded.