Complex Analysis

“ca: chapter_03” — 2012/3/27 — 20:49 — page 102 — #15

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Complex Analysis

Hence,

Log z = ln |z|+iArg z (3.21)

Now, equation (3.20) reduces to

log z = Log z + 2nπi, where n ∈ I

Equation (3.21) shows that Log z is a single-valued function and it coincides with natural logarithm

when z is a positive real number, i.e. Log x =lnx when z = x.

We know that arg z =Argz +2nπ . Thus, equation (3.20) can be rewritten as

log z = ln |z|+i arg z (3.22)

⇒ log z = ln r + iθ, where r =|z| and θ = arg z (3.23)

As we know from the properties of exponential function that

|e

z

|=e

x

and arg(e

z

) = y + 2nπ, where n ∈ I

∴ log(e

z

) = ln |e

z

|+i arg (e

z

) [Using (3.22)]

= ln



e

x



+ i

(

y