Complex Integration over a Contour
Buongiorno(/Bueno sera)!
Last post we discussed on the notions and some definitions of Complex Integration and arcs. Today we will move forward more toward the concept of Complex Integration over a Contour.
Contour Integration
Contour integration is a powerful technique, based on complex analysis, that allows us to solve certain integrals that are otherwise hard or impossible to solve. Contour integrals also have important applications in physics, particularly in the study of waves and oscillations.
Contour Integrals
Suppose f is a complex function of a complex variable. A straight-forward way to define the integral of f(z) is to adopt an analogous expression:
Since f takes complex inputs, the values of z_n need not lie along the real line. In general, the complex numbers $z_n$ form a set of points in the complex plane. To accommodate this, we can imagine chaining together a sequence of points
separate by displacements
Then the sum we are interested in is
In the limit
each displacement
becomes infinitesimal , and the sequence of points
becomes continuous trajectory in the complex plane . Such a trajectory is called a Contour. Let us denote a given contour by an abstract symbol, such as
. Then the Contour integral over Tau is defined as
The symbol $\tau$ in the subscript of the integral sign indicates that the integral takes place over the contour $\tau$. When defining a contour integral, it is always necessary to specify which contour we are integrating over. This is analogous to specifying the end-points of the interval over which to perform a definite real integral. In the complex case, the integration variable z lies in a two-dimensional plane (the complex plane), not a line; therefore we cannot just specify two end-points, and must specify an entire contour. Also, note that in defining a contour Γ we must specify not just a curve in the complex
plane, but also the direction along which to traverse the curve. If we integrate along the same curve in the opposite direction, the value of the contour integral switches sign (this is similar to swapping the end-points of a definite real integral)
Contour Integral along a Parametric Curve
Simple contour integrals can be calculated by parameterizing the contour. Consider a contour integral
where f is a complex function and $\tau$ is a given contour. we can describe a trajectory in the complex plane by a complex function of a real variable,
where
The real numbers $t_1$ and $t_2$ specify two complex numbers, $z(t_1)$ and $z(t_2)$, which are the end-points of the contour. The rest of the contour consists of the values of z(t) between those end-points. Provided we can parameterize Γ in such a manner, the complex displacement dz in the contour integral can be written as
Then we can express the contour integral over $\tau$ as a definite integral over
Sorry for such a late post. Uni life is hard i knew but Hostel screwed me. Made some memories and moments and lots of tension and stress owing to upcoming and ongoing midsems. But I’ll try to be more punctual and the conclusion to the Complex Integration is coming soon!
For 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
