Hans Georg Schaathun
September 2016
$$e^{a+iy}$$
$$\frac{d}{dz} e^{z} = e^{z}$$
$$z=x+iy$$
$$f(z) = e^x\cos y + ie^x\sin y$$
$$u(x,y) = \Re(f(x)) = e^x\cos y$$
$$v(x,y) = \Im(f(x)) = e^x\sin y$$
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} $$
$$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} $$
$$f(z) = u(x,y) + i\cdot v(x,y), \quad \quad z=x+iy$$
$$u(x,y) = \Re(f(x)) = e^x\cos y$$
$$v(x,y) = \Im(f(x)) = e^x\sin y$$
$$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = e^{x}\cos y + i\cdot e^x\sin y = f(z)$$