I am working with a unit circle with imaginary integration. I know from experience that this can be written as $f(\theta)=\cos t+ i \sin t$ or $e^{i \theta } $
My question would be if i have a circle with radius 2 and center 3, can i say this is $3+2e^{i \theta }$ and what would it be in terms of sin and cos?
A second question i have is how would i write a straight line from (0,0) to (1,5) in terms of imaginary co-ordinates?
Thank you very much, i hope all is clear
For your first question, the answer is yes; that is, a circle of radius $2$ and center $x = 3, y = 0$ can be parametrized by $3 + 2e^{i \theta}$, where $\theta$ goes from $0$ to $2 \pi$. As you know, $e^{i \theta} = \cos(\theta) + i \sin(\theta)$, so the circle is given by $3+2(\cos(\theta) + i \sin(\theta)) = 3 + 2 \cos(\theta) + 2i \sin(\theta)$. This is as simple as it gets.
Now, suppose you have two points in the complex plane, $w$ and $z$. Then the straight line from $w$ to $z$ is parametrized by $f(t) = t*z + (1-t)w$, where $t$ goes from $0$ to $1$. As you can see, $f(0) = w$ and $f(1) = z$. Hence, the straight line from $0 + 0i$ to $1 + 5i$ is given by plugging in $w = 0$ and $z = 1+5i$, giving us $f(t) = t(1+5i)$.