LOGARITHMIC FUNCTION OF A COMPLEX NUMBER

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Let z be a non-zero complex number. Then there always exists a complex number w such that  = z. w is said to be a Logarithm of z. 

Again  =   where n is an integer. This shows that if it is a logarithm of z, then  is also a logarithm of z. This means that 'logarithm of z' is a many-valued function of z. This is denoted by 

                                        Log z =  

Of the many values of the logarithm of z, a particular one is called the principal value and is denoted by log z. Since z is a non-zero complex number, z has a polar representation.

 Let z = r(cosθ+isinθ)   -π<θ≤π (a polar form with amp z). 

Let w=u+iv be a logarithm of z. Then  =z. This gives 

                                               (cos v+isin v) =r(cosθ+i sin θ) 

cos v= rcos θ     and        sin v = rsinθ 

We have      = r  and y cosv= cos θ, sin v= sin θ.

 These determine u = log r  and v=θ+2nπ, where n is an integer. 

              w= log r+i(θ+2nπ),-π<θ≤π

   i.e.    Log z = log r+i(θ+2nπ) = log|z|+i(arg z+2nπ). 

The principal value of Log z, denoted by log z, is the value corresponding to n=0. 

Thus,     log z = log|z|+i arg z .