# (Complex Variables) Differentiability of Arg z

1. Feb 20, 2006

### irony of truth

I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have

f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z.

(Note that /\ is the triangle symbol)
Also, let /\z = p + iq, where p and q are real values.

Arg z = Tan^-1 (y/x)...how will I continue from here?

2. Feb 23, 2006

### relinquished™

so you want to get

$$f'(z) = \lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}$$

Express the limit in terms of $$u(x_0,y_0) and v(x_0,y_0)$$, that is,$$x_0, y_0, \Delta x, \Delta y$$. then evaluate the limit using 2 approaches: when $$\Delta x = 0$$ and $$\Delta y = 0$$. If f(z) = Arg z is differentiable, the derivative should be equal in both cases.

Last edited: Feb 23, 2006