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What are Complex Variables?

Welcome back MechanicaLEi, did you know that many branches of physics like hydrodynamics, thermodynamics, quantum mechanics and by extension nuclear and aerospace engineering make use of complex variables? This makes us wonder, what are complex variables? Before we jump in check out the previous part of this series to learn about how Laplace transforms can be used to solve boundary value problems? Now, Complex variable, as the name suggests, is a variable that can take on the value of a complex number. In basic algebra, the variables x and y generally stand for values of real numbers. However, the algebra of complex numbers (called complex analysis) uses the complex variable z to represent a number of the form a + ib. The modulus of z is its absolute value and is given by root of a squared plus b squared. A complex variable may also be graphed as a vector from the origin to the point (a, b) in a rectangular coordinate system, its modulus corresponding to the vector’s length, called an Argand diagram, with a and b representing the real and imaginary part of the complex variable, respectively. There are three major functions of complex variables namely: Holomorphic function, Analytic function and Cauchy-Riemann equations. First let's look at the definition of Holomorphic function. Given a complex-valued function f of a single complex variable, the derivative of f at a point z0 in its domain is defined by the following limit. If the limit exists, we say that f is complex-differentiable at the point z not. If f is complex differentiable at every point z0 in an open set U, we say that f is holomorphic on U. Hence, we first saw what complex variables are and then went on to see what holomorphic functions are? In the next episode of MechanicaLEi find out what analytic functions are? Attributions: Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/...) Source: http://incompetech.com/music/royalty-... Artist: http://incompetech.com/ Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fab...) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/...) Source: http://freemusicarchive.org/music/Fab...