1. ## Parametrization

Let GAMMA be the path with parametrization

gamma(t) = 2 - i + 3e^it

Sketch GAMMA indicating the direction of increasing t and identifying it's inital and final points in cartesian form.

By applying geometric description of the function f(z)= 5i(z + i) to GAMMA, sketch the path f(GAMMA). Indicate the direction of f(GAMMA) and identify it's inital and final points in cartesian form.

Write down the standard parametrization of f(GAMMA)

2. Hi. Consider the real and imaginary parts of your gamma(t).

Re[Gamma(t)] = 2 + 3*Cos[t]
Im[Gamma(t)] = 3*Sin[t] - 1

So on the real/imaginary plane, your real (x) coordinate is given by:

2 + 3*Cos[t]

and your imaginary (y) coordinate is given by:

3*Sin[t] - 1

This will give a circle of radius 3 with center at (2,-1) that is 2*Pi periodic. Plug in t = 0 and it will give you your starting and ending point (ending being at t = 2*Pi) since they are the same point.

For the second part, it is transforming your gamma into something else. Put in your gamma for z and multiply it out and then once again separate the real and imaginary parts and you will have your x(t) and y(t).

As far as standard parametrization, I'm not sure what that is.

3. Cheers man, I had posted this question in amongst some others on another post called complex functions but noone had replied.
I had in that post that 3e^it is equal to 3(cost + isint)
That's where you are getting the real and imaginary parts from I think.
Right time to take another look at it, cheers again !