I Finding the inverse tangent of a complex number

AI Thread Summary
The discussion focuses on finding the arctangent of a complex number, expressed as w = arctan(z), where z = x + iy. A proposed formula involves logarithmic expressions, which the original poster wishes to avoid, seeking results in terms of trigonometric and hyperbolic functions instead. The conversation highlights the transformation of the problem into solving equations involving cosine and sine functions of complex arguments. By applying angle sum identities and recognizing hyperbolic relationships, the discussion derives equations relating the tangent and hyperbolic tangent of the angles. Ultimately, the goal is to express u and v as functions of x and y without relying on logarithmic forms.
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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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