
Complex Functions
Argh ok a couple of clues here would be useful !
Let f(z) = 5i(z + i)
1  Describe the effect f has on a typical point of C in terms of geometric transformations.
Is it that it is scaled by a factor of 5, moved up 1 unit and rotated by 90 degrees ?
2  Let GAMMA be the path with the parametrization
gamma(t) = 2  i + 3e^it
Sketch GAMMA indicating the direction of increasing t and identifying
its initial and final points in cartesian form.
So I change 3e^it to 3(cost + isint) ?
So I get 2i+3(cost + isint)
Then do i need to get the real and imaginary parts for the parametric equations I think ?
3  By applying your geometric description of the function f to GAMMA, sketch the path f(GAMMA). Indicate the direction of f(GAMMA) and identify it's inital and final points in Cartesian form.
4  Write down the standard parametrization of f(GAMMA)
Bit stuck on 3 and 4.
5  Let f(z) = exp(12iz)
g(z) = Log(12iz)
h(z) = (12iz)^3i
a  write down the domain of each of the functions f,g and h
f would be all complex numbers.
In g the part 12iz would have to be positive but I'm not sure how I describe that.
Similar case for h
b  Explain why f does not have an inverse fucntion.
Because of the periodicity e^z+2piei = e^z
c  Show that g has an inverse function and find it's domain and rule
erm ?!?
Any clues pointers appreciated cheers ! :)