Hey,

I just wanted to see if i was approaching these correctly, The final two are what i'm struggling with at the moment and I have a little confusion with if cauchys theorem works backwards.

a) |2+3i|=|2-3i| so false

b) True since when the coefficients are real the roots come in complex conjugate pairs

c) Using triangle inequality, 1/|z^2+1|=> 1/|z^2| +2 = 1/ (x^2+y^2+2)

1/ (x^2+y^2+2)<=1/x^2+2<=1/2

d)False, cannot be every f since f must be analytic within the domain and curve region

e) False, i think, by Cauchys integral formula C must be a simple closed curve enclosing Zo, so as C in this question is just a line, False

f) I think this is true, but i'm not sure if you can use cauchys theorem backwards. So true as int f(z)dz=0 if f is analytic in C. So since int f(z)dz=0, f must be analytic inside and on C

g) and h) I'm not sure how to approach this, but I was thinking you could just let an be somthing like 1/sqrt(3)^n for the h), then i think it would work, so h) true

but for g) i cant see a way to do somthing similar, so i think the answer is false

Does my working seem correct?

Thanks in advance