\[i^2=-1, or, i=\sqrt{-1} .\]

Everything starts from here! The imaginary number $i$ is defined as the square root of -1, which is not a real number.

Complex analysis can be traced back to the 19th century or even earlier. An important part of it, conformal mappings, has many applications in physics, engineering. More recently, complex dynamics, fractals generated more interests for the classical subject. Another important application of complex analysis today is in string theory in physics.

Even more recent, Schramm–Loewner evolution, a new field in probability theory, emerged around 2000 that uses complex analysis, classical differential equation theory and probability theory to study some random processes on the plane.

As an example, have you ever computed the indefinite complex integral

\[\int\frac{1}{1+z^2}dz ?\]

Of course, you know the answer

\[\int\frac{1}{1+z^2}dz =\tan^{-1}z+C,\]

but have you written out all the details and really understand everything? Have you used at least 5 different ways to calculate the real integral below?

\[\int_{-\infty}^{+\infty}\frac{1}{1+x^2}dx \]

An example in the demo course: http://www.mathwit.com/math/m2/mod/resource/view.php?id=163

Here at MathWit, we can help you understand complex analysis, from undergraduate course to some graduate courses. Topics covers Fourier transform, using contour integral methods to compute integrals, connections with probability theory and more. We can possibly discuss some of your research topics for PH.D students and post-docs.