analytic geometrycomplex numberscomplex-analysis
I'm confused about the straight line equation in complex plane: how does
$0 = Re((m+i)z + b)$ come from $y = mx + b$?
I mean when I see $y = mx + b$, I can draw a graph in my mind, but when I see $0 = Re((m+i)z + b)$, there is nothing on my mind.
How can I connect the two equatinos?
Does anyone could help me, thanks!
You can use the Open Mapping Theorem, which states:
Any non-constant holomorphic function that is defined on a non-empty open subset $ U $ of $ \mathbb{C} $ must map open subsets of $ U $ to open subsets of $ \mathbb{C} $.
As $ f[\mathbb{C}] $ is forced to be contained in a straight line, and as a straight line cannot contain a non-empty open subset of $ \mathbb{C} $, it follows that $ f $ must be a constant function. The correct answer is therefore (b).
$b$ represents the y-intercept of the line: where the line crosses the y-axis.
It can be found by setting $x = 0$ (the line crosses the y-axis when and only when $x = 0.$)
An example of such a line, $$y = 3x + \underbrace{7}_{\large b}$$ crosses the y-axis at the point $(0, 7)$.
When $x = 0$, you see that $y = 7 = b$.
Best Answer
Let $z=x+iy$. Then, $(m+i)(x+iy)+b=mx+ix+imy-y+b$. Thus, looking at the real part, we get $mx-y+b=0$, also known as $y=mx+b$.