Analytic Function

The theory of complex functions mainly deal with analytical functions.

\(f'(z) = \frac{d}{dz}f(z) = \lim_{\Delta z\to 0}\frac{f(z+\Delta z) - f(z)}{(z+\Delta z) - z}= \lim_{\Delta z\to 0}\frac{f(z+\Delta z) - f(z)}{\Delta z}\)

Let \(\Delta z = \Delta x\). \(f'(z) = \lim_{\Delta x\to 0}\frac{f(z+\Delta x) - f(z)}{\Delta x} = \lim_{\Delta x\to 0}\frac{u(x+\Delta x,y) + iv(x+\Delta x,y) - u(x,y) -iv(x,y)}{\Delta x} = \lim_{\Delta x\to 0}\frac{[u(x+\Delta x,y) - u(x,y)] + i[v(x+\Delta x,y) -v(x,y)]}{\Delta x} = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}\)

Let \(\Delta z = i\Delta y\). \(f'(z) = \lim_{\Delta y\to 0}\frac{f(z+i\Delta y) - f(z)}{i\Delta y} = \lim_{\Delta y\to 0}\frac{u(x,y+\Delta y) + iv(x,y+\Delta y) - u(x,y) -iv(x,y)}{i\Delta y} = \lim_{\Delta y\to 0}\frac{[u(x,y+\Delta y) - u(x,y)] + i[v(x,y+\Delta y) -v(x,y)]}{i\Delta y} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}\)

\(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}\).

Definition

A function \(f\) is analytic if \(f\) has a derivative at \(z\) and in the neighborhood of \(z\). (I.e. in a region \(\mathcal{R}\subset\mathbb{C}\))

Also called entire or holomorphic.

Entire means that \(f\) is holomorphic in \(\mathcal{R}=\mathbb{C}\).

Meromorphic means that \(f\) is holomorphic in \(\mathcal{R}=\mathbb{C}\) except for a set of isolated poles.

Cauchy-Riemann Conditions

Thus, \(\frac{\partial u}{\partial x} = i\frac{\partial v}{\partial y}\) and \(-\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}\) This is called the Cauchy-Riemann Conditions.

Aside

If you can write a function \(f\) solely in terms of \(z\) (not \(z^*\)) (e.g. \(f(z) = 1/z,\sin z,z^2\)) then it is differentiable everywhere it is finite. An example of one that is not, \(|z|^2 = zz^2\) is not entire.

If it is differentiable then all of the derivative rules from before hold. I.e. \(\frac{d}{dz}z^n = nz^{n-1}\) etc.