Analysis is the branch of mathematics concerned with limit processes and therefore provides the foundation for a rigorous study of concepts like the infinitesimally small and the infinitely large. Based on the ideas of Analysis one can introduce the notions of convergence and divergence and hence study trends in sequences of numbers and data. For example, it is remarkable that the infinite sequence of inverse squares of natural numbers sums up to a finite quantity, in fact a transcendental number.
This module introduces you to the concepts of convergence and divergence and explains how these ideas can be rigorously applied to define the rate of change of a function (derivative) or to generalise the technique of summation to functions (integral). The module also extends these concepts from real numbers to complex numbers, resulting in a beautiful theory with very powerful results.
Visualising a complex function would usually involve four dimensions, but plotting its real and imaginary parts individually already reveals highly valuable information.
Note how the graph of the complex square root function (left) penetrates itself so that paths around the centre can lead you back to where you started, whereas the graph of the complex logarithm function (right) spirals around the vertical axis.
