Find the principle value of the argument of the complex number $$z=a+bi$$ $$(a,b\in R, a^2+b^2
e 0)$$

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Arg $$(z)$$ is the argument which lies in the interval (−π,π]. Depending on different values of $$a$$, and $$b$$, there are 8 cases. Case 1, if $$a>0, b>0$$ when the Arg $$(z)$$ lies in the first quadrant. From the definition of the argument, $$ an heta =\dfrac{b}{a}$$, then the principle value of the argument is the inverse tangent function, that is, $$ ext{Arg}(z) = \arctan\dfrac{b}{a}$$c Case 2, if $$a<0, b>0$$ when the point represented by the complex number lies in the second quadrant. $$ ext{Arg}(z) =\pi- \arctan\dfrac{b}{|a|}$$c Case 3, if $$a<0, b<0$$ when the point represented by the complex number lies in the quadrant III. $$ ext{Arg}(z) =-\pi+ \arctan\dfrac{|b|}{|a|}$$c Case 4, if $$a>0, b<0$$ when the $$ ext{Arg}(z)$$ lies in the fourth quadrant. $$ ext{Arg}(z) = -\arctan\dfrac{|b|}{a}$$c Case 5, if $$a=0, b>0$$ when the $$ ext{Arg}(z)$$ lies on the positive side of the imaginary axis $$ ext{Arg}(z)=\dfrac{\pi}{2} $$c Case 6, if $$a=0, b<0$$ when the $$ ext{Arg}(z)$$ lies on the negative side of the imaginary axis $$ ext{Arg}(z)=-\dfrac{\pi}{2} $$c Case 7 if $$b=0, a>0$$ when the $$ ext{Arg}(z)$$ lies on the positive side of the real axis $$ ext{Arg}(z)=0 $$c Case 8 if $$b=0, a<0$$ when the $$ ext{Arg}(z)$$ lies on the negative side of the real axis $$ ext{Arg}(z)=\pi $$c

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