In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function x ↦ f(x)
to its translation x ↦ f(x + a). In time series analysis, the shift operator is called the lag operator.
Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive definite functions, derivatives, and convolution....
Let $f:\Bbb{C}\setminus(-\infty, 0]\rightarrow \Bbb{C}$ where $f(z)=z^{1+i}:=e^{(1+i)\log(z)}$. We consider $$S=\{0<Arg(z)<\pi/6\}$$
I want to draw $f(S)$.
If I take $z\in S$ then $z=re^{it}$ where $r\in [0,\infty)$ and $t\in (0,\pi/6)$. Then $\log(z)=\log(r)+it$ where now $\log(r)\in (-\infty...
