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BodhiAI
Nov. 11, 2019

Square root of complex number and logarithm of complex num.:

Square Root of A Complex Number

Let z = a+ be any complex number such that a, b

Let where x, y R

Squaring both sides, we have 

a+ = (x+y)2 

According to Law of Equality of Complex Numbers, we have .....(1)

.....(2)

and 2xy = b

Now,

Adding (1) & (3), we have 

......(4)

Now, {using (2)}

Now, if b > 0, then

and if b < 0, then

If b < 0, then

b < 0 and according to rule two negatives can never rest under a single root. 

Case I: b > 0

........(5)

{using (4) & (5)}

Problem Solving Trick

where b > 0

where Im (z) > 0

Also, where b > 0