Sneak peek into the Integrals of Complex functions

An integration of a function $f(z)$ over $z$ is defined as, 

$$I = \int^{b}_{a} f(z)dz$$

But we can convert this integral into a summation and write,

$$I = \sum^{n}_{k=1} f(\xi_k)(z_k - z_{k-1})$$

where, $z_0 = a$ and $z_n = b$. So the above expression becomes,

$$ I = \sum^{n}_{k=1}f(\xi_k)\Delta z_k$$

Now, let's talk about complex functions. So let's define a complex function $f(z)$, such that

$$f(z) \equiv u(x,y) + i v(x,y)$$

where, $z = x + i y$, So the equation follows, 

$$f(z)dz  = (u + i v)(dx + i dy)$$

$$f(z)dz = (udx - vdy) + i (vdx + udy)$$

Now, let's substitute some terms to simplify the equation. The following substitutions are made,

$\overrightarrow{A}(\overrightarrow{r}) = (udx - vdy) $ and $\overrightarrow{B}(\overrightarrow{r}) =  (vdx + udy)$

Now, having this equations and substitution in hand, we now integrate the above relation,

$$\int^{b}_{a}f(z)dz = \int^{b}_{a}(udx - vdy) + i \int^{b}_{a}(vdx + udy)$$

$$\int^{b}_{a}f(z)dz = \int^{b}_{a}\overrightarrow{A}(\overrightarrow{r}) + i \int^{b}_{a}\overrightarrow{B}(\overrightarrow{r})$$

Now, that being defined let's look into some theorems.

Jordan Curve Theorem Every non-intersecting curve divides the complex plane into an interior and an exterior. 

[Note: Interior and Exterior are just the concept of being inside or outside of a region defined by the curve or the function defined.]

Cauchy-Gowsal Theorem If $f(z)$ is analytic in a region $R$ and if it's boundary is $C$, then 

$$\oint_C f(z) dz = 0$$

Monera's Theorem If $f(z)$ is continuous in $R$ and $$\oint_C f(z) dz = 0$$ for every closed curve in $R$, then $f(z)$ is analytic in $R$.

Ok, so that's a little sneak peek into the basic of integrating a complex function. So, stay tune for more exciting materials.