I Finding the inverse tangent of a complex number

bsaucer
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Inverse Tangent of complex number in rectangular form.
Let z=x+iy, and w=u+iv. I am looking for a formula to find the arctangent of z, or w=arctan(z). I want the results of u and v to be in terms of trigonometric and hyperbolic functions (and their inverses) and not in terms of logarithms. The values u and v should be functions of x and y.
 
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The formula
\tan^{-1}z=\frac{i}{2}\log\frac{i+z}{i-z}
seems useful to me but you do not like logarithm. 
 
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