# Complex calculus

### Defining complex valued functions

We can consider complex valued functions as a type of vector fields.

### Line integral of the complex plane

$$\int_C f(r) ds=\lim_{\Delta s rightarrow 0 }\sum_{i=0}^n f(r(t_i))\Delta s_i$$

$$\int_C f(r) ds=\lim_{\Delta s rightarrow 0 }\sum_{i=0}^n f(r(t_i))\dfrac{\delta r(t_i)}{\delta t}\delta r_i$$

$$\int_C f(z) dz=\int_a^b f(r(t_i))\dfrac{\delta r(t_i)}{\delta t}\delta r_i$$

### Wirtinger derivatives

Previously we had partial differentiation on the real line. We could use the partial differention operator

We want to find a similar operator for the complex plane.

### Line integral of the complex plane

$$\int_C f(r) ds=\lim_{\Delta s rightarrow 0 }\sum_{i=0}^n f(r(t_i))\Delta s_i$$

$$\int_C f(r) ds=\lim_{\Delta s rightarrow 0 }\sum_{i=0}^n f(r(t_i))\dfrac{\delta r(t_i)}{\delta t}\delta r_i$$

$$\int_C f(z) dz=\int_a^b f(r(t_i))\dfrac{\delta r(t_i)}{\delta t}\delta r_i$$

### Complex smooth functions

If a function is complex differentiable, it is smooth.

### Cauchy-Riemann equations

Consider complex number z=x+iy

A function on this gives:

$$f(z)=u+iv$$

Take the total differential of :

$$df/dz=\dfrac{\delta f}{\delta z}+\dfrac{\delta f}{\delta x}\dfrac{dx}{dz}+\dfrac{\delta f}{\delta y}\dfrac{dy}{dz}$$

We know that:

• $$\dfrac{dx}{dz}=1$$

• $$\dfrac{dy}{dz}=-i$$

We can see from this that

• $$\dfrac{du}{dx}=\dfrac{dv}{dy}$$

• $$\dfrac{du}{dy}=-\dfrac{dv}{dx}$$

These are the Cauchy-Riemann equations