 Welcome back MechanicalEI. Did you know that 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 lapless 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 plus i b. 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 comma b in a rectangular coordinate system. It's modulus corresponding to the vector's length called an argon diagram with a and b representing the real and imaginary part of the complex variable respectively. There are three major functions of a complex variable, namely holomorphic function, analytic function and Cauchy-Rehmann 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 z naught in its domain is defined by the following limit. If the limit exists, we say that f is a complex differentiable at point z naught. If f is complex differentiable at any point z naught 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. Like, subscribe and comment with your feedback to help us make better videos. Thanks for watching.