Complex Analysis

[Guille]

I wonder a lot about this branch of mathematics, probably because it's dificult (I like mathematical chalenges) and because I don't really understand it. I have a series of questions which I will ponder in this thread. The first one is:

What are the applications that can be done for the practical use of complex analysis?

[AntonioLao]

What are the applications that can be done for the practical use of complex analysis?

It started when mathematicians were looking for the roots of the polynimial equation x²+1=0. the roots are +i and -i and i=√-1. The result of these search is the creation of a new complex number system z=a+bi where a is the real part and bi is the imaginary part. It becomes complex analysis when calculus is applied to these complex numbers.

[Guille]

I understand complex numbers, but what do you mean with the application of calculus to these?

[AntonioLao]

When differential calculus is applied to complex numbers, their derivatives exist and can be manipulated and other properties can be found. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, 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 that 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.

[Guille]

Thanks for the info. What different kinds of complex analysis exist? What's their difference?

[AntonioLao]

What different kinds of complex analysis exist?

I think it's hypercomplex analyses and their algebras. For examples: vector analysis, tensor analysis, quaternion analysis, octanion analysis, etc.

[Guille]

I think it's hypercomplex analyses and their algebras. For examples: vector analysis, tensor analysis, quaternion analysis, octanion analysis, etc.

I remeber discussing with you on vector and tensor analysis, and that tensor analysis wasn't very onventional to you.

I know that quaternions where first developed by Hamilton, but I don't know what they are all about?

[AntonioLao]

I know that quaternions where first developed by Hamilton, but I don't know what they are all about?

Hamilton was looking for 3D analogue of the complex numbers. So in addition to i, he added j and k.

[Guille]

Hamilton was looking for 3D analogue of the complex numbers. So in addition to i, he added j and k.

I don't really understand. What does "3D analogue of the complex numbers" mean?

Are j and k numbers out of the complex plane, like, they form a nex dimension in the graph?

[AntonioLao]

I don't really understand. What does "3D analogue of the complex numbers" mean?

the complex plane is partitioned by two axes, the real and the imaginary axis. In the quaternion space, it is partitioned by 4 axes, the i-axis, the j-axis, and the k-axis and of course the real usual real axis. The differences are the different ways of doing addition, subtraction, and multiplication. I am not sure whether quaternion can do division. Nevertheless, quaternion is really composed of a vector part and a scalar part. Later, Heaviside was able to separate them into just vectors and scalars.