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Find the domain of analyticity for f(z)={\mathrm  {Log}}(3z-i)\,

The principal valued {\mathrm  {Log}}\, is analytic everywhere except the ray in the complex plane x\leq 0,y=0\,. In this case,

3z-i=3(x+iy)-i=3x+i(3y-1)\, so this function is not analytic when x\leq 0\, and y=1/3\,.

Now, in this slit plane the derivative can be computed:

f'(z)={\frac  {1}{3z-i}}{\frac  {d}{dz}}(3z-i)={\frac  {3}{3z-i}}\,

Main Page : Complex Variables : Exponential and Log