 Welcome back MechanicalEI! Did you know that Laplace transformations were bought into popularity after its use during World War II? This makes us wonder, what are Laplace transformations? Before we jump in, check out the previous part of this series to learn about what Euler's column theory is. Now, the Laplace transform of a function f of t defined for all real numbers t is the function f of s, which is a unilateral transformation defined by f of s equals integral of f of t into e power minus st with respect to dt from 0 to infinity, where s is a complex number frequency parameter and is equal to sigma plus iota omega with real numbers sigma and omega. An alternate notation for the Laplace transformation is l of f instead of f of t. The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on closed interval of 0 to open interval of infinity. While studying Laplace transforms, it is also important to know the definition of bounded variables. In mathematical analysis, a function of bounded variation also known as bv function is a real valued function whose total variation is bounded, finite. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of y-axis neglecting the contribution of motion along x-axis traveled by a point moving along the graph has a finite value. Hence, we first saw what Laplace transforms are and then went on to see what functions of bounded variation are. So like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also, thanks a lot for those constructive comments. You helped the channel grow. So, here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what linearity property is.