Complex Variable Question?
Hey people! just wonderin could anyone out there help me with this question, having some difficulty...
Find all values of
log(1 + i)
and determine the principal value Log(1+i). Hence fi nd the real and the
imaginary parts of the principal value of
(1 + i)^1+i
any help would be much appreciated!
Update:Thanks kb that really helps and gives me some insight! But isnt the principal value now when k=0? So the principal value of log(1+i) should be (1/2)ln2+pi i/4? I could be wrong, really didnt have a great professor for this so ended up havin to try and learn my whole course from the net!
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Verified answer
log(1 + i) = Log(1 + i) + 2πki, where k is any integer
..............= [ln |1 + i| + i Arg(1 + i)] + 2πki
..............= [ln √2 + i * π/4] + 2πki
..............= (1/2) ln 2 + (πi/4) (1 + 8k), for any integer k.
Therefore, (1 + i)^(1 + i)
= exp [log ((1 + i)^(1 + i))]
= exp [(1 + i) log(1 + i)]
= exp [(1 + i) * ((1/2) ln 2 + (πi/4) (1 + 8k))]
= exp {[(1/2) ln 2 - (π/4) (1 + 8k)] + i [(1/2) ln 2 + (π/4) (1 + 8k)]}
= exp [(1/2) ln 2 - (π/4) (1 + 8k)] * exp {i [(1/2) ln 2 + (π/4) (1 + 8k)]}
= √2 exp [(-π/4)(1 + 8k)] * {cos [(1/2) ln 2 + (π/4) (1 + 8k)] + i sin [(1/2) ln 2 + (π/4) (1 + 8k)]}
= √2 exp [(-π/4)(1 + 8k)] * {cos((1/2) ln 2 + π/4) + i sin((1/2) ln 2 + π/4)}, for any integer k
since sine and cosine have period 2π.
I hope this helps!