you can't just SAY H is a homomorphism, you have to offer some evidence.

if , to show H is a homomorphism, you need to show that:

for all .

now (you should finish this....)

one way to write is . it should be clear that H cannot possibly be onto, because any image H(z) is always a positive real number. so Im(H) is certainly NOT C*.

also, ker(H) is NOT {1}, and certainly not R+. what is H(2)? what is H(-1), or H(i)? even better, what is H(√2/2 + i√2/2)? what does it mean to say ?

the kernel of H has an easy to remember shape.