PHYS20672 Complex Variables and Integral Transforms

Aims:
To introduce students to complex variable theory and some of its many applications.

Learning Outcomes:
On completion successful students will:


 * 1) have an understanding of complex variable theory and know how this is used to evaluate certain integrals using the residue theorem.
 * 2) be able to solve certain linear ordinary and partial differential equations by Laplace and Fourier Transforms.
 * 3) be able to carry out substantial calculations involving the topics in the syllabus and recognise the techniques necessary.

Complex Numbers
Complex numbers are of the form
 * $$z=x+iy\,$$ where $$x,y \in \mathbb{R}$$.

They can be represented on a 2D plane, called the complex plane or an Argand Diagram. In this representation, x and y take on their respective co-ordinates in the plane and can therefore be redefined in terms of planar polar co-ordinates,


 * $$r^2 = x^2 + y^2\,$$,

and


 * $$\tan\theta = \frac{y}{x}\,$$.

Using these and Euler's equation it can be found that


 * $$x+iy = r(\cos\theta + i\sin\theta) = re^{i\theta} \,$$,

where $$\theta$$ is only defined from $$0$$ to $$2\pi$$.