What are Taylor’s and Laurent’s series?
Channel: MechanicaLEi
Date: 2018-08-13

Welcome back MechanicaLEi, did you know that a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point and that a Laurent series of a complex function f of z is a representation of that function as a power series which includes terms of negative degree.? This makes us wonder, what are Taylor’s and Laurent’s series? Before we jump in check out the previous part of this series to learn about what properties of line integral are? Now, The Taylor series of a real or complex-valued function f?of x that is infinitely differentiable at a real or complex number a is the power series, which can be written in the more compact sigma notation as summation of f power n of a upon n factorial into x minus a, the whole raised to n, from n equals to zero to infinity. Here, f power n of a denotes the nth derivative of f evaluated at a. The Laurent series of a complex function f of z is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series for a complex function f of z about a point c is given by f of z equals to summation of a n into z minus c raised to n from n equals to minus infinity to infinity, where a n and c are constants defined by a line integral which is a generalization of Cauchy's integral formula. Hence, we first saw what Taylor's Series is and then went on to see what Laurent's Series is?
In the next episode of MechanicaLEi find out what Residue theorem is?

Attributions:

Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/)
Source: http://incompetech.com/music/royalty-free/index.html?isrc=USUAN1100255
Artist: http://incompetech.com/

Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fabian_Measures/) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/by/4.0/)
Source: http://freemusicarchive.org/music/Fabian_Measures/Emotion_Music/Subtle_Library