Touching Complex Integration

“Agriculture can only thrive on rich soil” — Sadhguru

Before we start talking about complex integration we need to be aware of the notion of Complex Integration. The essence to why is not so easy to be covered yet not so hard so I believe this post will be sufficient.
The concept of definite integral of a function of a real variable does not extend to the domain of complex variables. But in case of complex function f(z) the path of the definite integral

can be along any curve from z=a to z=b; so it’s value depends upon the path(curve) of integration. This variation in values can be made to disappear if the different curves(paths) from a to b are regular curves.

Now that our premise is set we will discuss Some Important Definitions,

Partition

Let [a,b] be a closed interval where a,b are real numbers. Then the set of points P=t0,t1,…,tnt0,t1,…,tn{t_0,t_1,…,t_n} where a=t0<t1<t2…<tn=ba=t0<t1<t2…<tn=ba=t_0<t_1<t_2…<t_n=b is called the Partition of the interval [a,b]. The greatest of the numbers t1 — t0,t2 — t1,…,tn — tn−1t1 — t0,t2 — t1,…,tn — tn−1t_1 — t_0, t_2 — t_1,…,t_n — t_{n-1} is called the Norm of the Partition P and denoted by ||P||.

Continuous Arc

If a point z on an arc is such that

then we may write x=Φ(t) and y= ψ(t)

If Φ(t) and ψ(t) are real continuous functions of the real variable t defined in the range

then the arc is called Continuous Arc.

Multiple Point:

If the equation

or

are satisfied by more that one values of t in the given range, then the point z, or say the point (x,y) is called Multiple Point of the arc

Jordan Arc

A Continuous Arc without Multiple Points is called a Jordan Arc.

A Continuous Jordan Arc consists of a chain of continuous arcs. And by Contour we mean a Jordan Curve consisting of continuous chain of a finite number of regular arcs.

If A be a starting point of the first arc and B be end point of the last arc, the the integral along such a curve is written as

If the starting point A of the arc coincides with the end point B of the last arc then the contour AB is said to be closed. The integral along such closed contour is written as

and is read as integral f(z) taken over the closed contour C. Although

doesn't indicate the direction along the curve, but it is conventional to take the direction positive which is anti-clockwise unless indicated otherwise.

Now that the table is set we can nonchalantly discuss complex integration over a contour. I plan to do that in the next post. Hope this was helpful!

Reference

Complex Analysis by Kedar Nath Ram Nath Publication
Contour Integration (Complex Methods for the Sciences) by Y.D Chong (2020)
Elements of Complex Analysis by Shobhakar Ganguly
https://en.m.wikipedia.org/wiki/Contour-integration

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