Taylor Functions – explanation and software tools

January 7, 2019
Bill Toulas

Taylor functions are used in areas such as power engineering, fluid mechanics and bathymetry to simplify often complex functions into more understandable terms.
Manually calculating Taylor functions is mind blowing - thankfully there are an array of software tools to do this.
Where some exact elements of an equation are not available, Taylor functions allow an improvement on traditional approximation calculations.

The Taylor Functions are series of sum terms that concern the values of the derivatives of a given function at a given point, so they are a part of Calculus. They are a clever practical way to expand and simplify complicated and transcendental functions into computable polynomials. The number of the sum terms that are used to represent the specific function is theoretically infinite, so we are either following an “adequate approximation” approach or we use software to calculate the series of the sum using a large number of terms.

Taylor Functions in practice

Here is an example to help you understand Taylor functions:

Let’s suppose that we want to calculate the function e^x for x=2. Euler’s number is an irrational number which means that its digits are infinite, so our calculation will be approaching a value of 2.718² = 7.387524.

Now if we use Taylor’s Functions, we first consider the derivative of e^x which happens to be e^x. So, for our series, we will get: “Σ _{(from 0 to n)} xⁿ/n!” which translates to: (x⁰)+(x¹/1!) + (x²/2!) + (x³/3!) +… (xⁿ/n!). For practical purposes, I will showcase the calculation of the series for the first five terms which are: (2⁰)+(2¹/1)+[2²/(1+2)]+[2³/(1+2+3)]+[2⁴/(1+2+3+4)]+[2⁵/(1+2+3+4+5)] = 1+2+2+1.333+0.667+0.267 = 7.267. Now it is obvious that this value is not close enough to e², but this is because we haven’t used enough terms in our series of sums. So, if we continue calculating the sum to up to 10 terms the value becomes 7.3889, and once we get to 20 terms it becomes 7.389056.

What is their use in Engineering

Most real things and systems in nature, engineering practice and phenomena are complex, at least to our mind. This means that their mathematical representation is equally complex making things harder for us to calculate. Taylor Functions are used exactly for the purpose of simplifying things to make them easier to work with. Here are a couple of examples of where Taylor Functions are used:

In power engineering, the flow of electric power in an interconnected system involves many variables that are initially unknown. To address this, engineers use the Newton-Raphson solution method that is based on the use of Taylor Functions.
Fluid mechanics engineers use the Taylor series in conjunction with the Navier-Stokes equation to achieve an accurate calculation method when studying arbitrary shapes with the Galerkin Computational method.
Similarly, bathymetry studies involve Taylor Functions based on the Dirichlet-Neumann operator to represent three-dimensional wave motion in two-dimensional mathematics (Hamiltonian).
Taylor functions are also used in error propagation studies as an alternative to the Monte-Carlo method.
Harmonic oscillation systems can be analysed more simply and essentially better understood through the use of Taylor Functions.
Engineering and physics simulation software such as ANSYS, MATLAB, and Abaqus use Taylor functions to calculate heat and mass transfer, fluid dynamics, etc.

Software tools

Because calculating Taylor Functions is only for psychopaths, doing so on a software tool is standard practice for an engineer nowadays. Here are some common online and offline tools that you can use for the calculation of Taylor Functions:

Online