# Complex calculus

### Complex-valued functions

### Defining complex valued functions

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

### Line integral of the complex plane

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

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

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

### Complex continuous functions

### Open regions

### Analytic continuation

### Analytic functions

### Circle of convergence

### Complex differentiation

### 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 integration

### Complex smooth functions

If a function is complex differentiable, it is smooth.

### All differentiable complex functions are smooth

### All smooth complex functions are analytic

### Singularities

### Contour integration

### Line integral

### Cauchy’s integral theorem

### 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

These are the Cauchy-Riemann equations