The application of these methods to real world problems include propagation of acoustic waves relevant for the design of jet engines, development of boundary-integral techniques useful for solution of many problems arising in solid and fluid mechanics as well as conformal geometry in imaging, shape analysis and computer vision.
This knowledge exchange day was part of a four month research programme at the Isaac Newton Institute on Complex Analysis: techniques applications and computations. It formed day three of the week long workshop Complex analysis in mathematical physics and applications.
The research programme brings together researchers from mathematics, physics and engineering communities, whose research shares a common theme of using complex analysis to attack real-world problems. This knowledge exchange event showcased the state of the art application of complex analysis methods to solve industrial driven problems, as well as where mathematical advances in this area are most needed.
Another key aim was to identify techniques most commonly used by end-users and where further improvements would be most beneficial. This will help focus the areas of investigation undertaken in the remainder of the research programme.
The programme for the day reflected the breadth of application areas where complex analysis methods are important and included talks representing both academic research and end-user perspectives from a range of different industrial areas. These talks also highlighted how complex analysis methods have the potential to tackle challenging problems in a number of areas including understanding of aeroacoustics, medical imaging methods, tissue engineering approaches and radar signal processing.
Furthermore, the increasing difficulty of many problems in these fields will help inform the agenda for complex analysis research. The four sessions that were covered are:. This event brought together mathematicians and scientists working at the forefront of complex variable theory and their applications, with end users from industry to further investigate opportunities for the use of complex variable methods in the solution of applied problems. Contour integration , for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.
The key result in complex analysis is the Cauchy integral theorem , which is the reason that single-variable complex analysis has so many nice results. A single example of the unexpected power of complex analysis is Picard's great theorem , which states that an analytic function assumes every complex number , with possibly one exception, infinitely often in any neighborhood of an essential singularity!
A fundamental result of complex analysis is the Cauchy-Riemann equations , which give the conditions a function must satisfy in order for a complex generalization of the derivative , the so-called complex derivative , to exist. When the complex derivative is defined "everywhere," the function is said to be analytic. Arfken, G. Orlando, FL: Academic Press, pp.
Boas, R. Invitation to Complex Analysis. New York: Random House, Churchill, R. Complex Variables and Applications, 6th ed. New York: McGraw-Hill, Conway, J.
Functions of One Complex Variable, 2nd ed. New York: Springer-Verlag, Forsyth, A. Theory of Functions of a Complex Variable, 3rd ed. Cambridge, England: Cambridge University Press, Ready to get started? Skip ahead to the Table of Contents.
This book is an interactive introduction to the theory and applications of complex functions from a visual point of view. However, it does not cover all the topics of a standard course. In fact, it is a collection of selected topics and interactive applets that can be used as a supplementary learning resource by anyone interested in learning this fascinating branch of mathematics.
Some of the topics covered here are basic arithmetic of complex numbers, complex functions, Riemann surfaces, limits, derivatives, domain coloring, analytic landscapes and some applications of conformal mappings.
What distinguishes this online book from other traditional texts in the first instance is the use of interactive applets that allow you to explore properties of complex numbers geometrically and analyze complex functions by using different techniques to visualize them. For the design of applets I used the following open-source softwares:. Although I advocate for the use of computers as an aid to geometric reasoning, I highly encourage you to practice your problem solving skills by solving the suggested exercises, or filling the missing details, that you will encounter throughout the sections.
Think of the computer as a physicist would his laboratory.
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