I have the following definition of the complex directional derivative :

$\displaystyle D_{w_1} f(z) = lim_{t \rightarrow 0} \frac{f(z+tw_1)-f(z)}{t} $ where $\displaystyle |w_1| = 1 $.

$\displaystyle D_{w_1} f(z) = lim_{t \rightarrow 0} \frac{u(x+tx_1,y+ty_1)-u(x,y)}{t}+i\frac{v(x+tx_1,y+ty_1)-v(x,y)}{t}$

From the definition of real directional derivative

$\displaystyle D_{w_1} f(z) = x_1\frac{du}{dx}+y_1\frac{du}{dy}+i(x_1\frac{dv}{d x}+y_1\frac{dv}{dy})$

But now if $\displaystyle w_1 = i $, $\displaystyle x_1 = 0 $, $\displaystyle y_1 = 1 $ and $\displaystyle D_{w_1} f(z) = \frac{du}{dy}+i \frac{dv}{dy}$ which does not agree with Cauchy Riemann equations.

Could someone tell me where I am mistaken please?