Taylor and Laurent Series

Taylor series can be expressed as an infinite summation of a function and this summation introduces the derivatives of the function in a single point. On the other hand, Laurent Series introduces a complex function as a power series. Laurent series can be applied to introduce the complex functions where the application of the Taylor series cannot be done. The principal part of the Laurent series is the part of the series with negative powers of z-z0. 

Laurent series

Pierre Alphonse Laurent developed the Laurent Series in 1843. Karl Weierstrass developed this series in 1841 but that article could not be published until after the death of Karl Weierstrass. Laurent Series of a complex function f (z) about a point c can be expressed as:

“ f (z) = n=-∞∑∞ an (z-c)n”

In the above equation, the an and c both are constant. Here “an” can be defined by the line integral and concludes with Cauchy’s integral formula. As a reason, here “an” can be written as: an = (1/2 ℼi) ∲ f(z) / (z-c)n+1 dz

The integration path is counterclockwise around the Jordan curve. This curve is enclosed by and lying in an annulus A. In this annulus f (z) is a holomorphic function. The derivation of this function can be validated in the annulus A. Laurent series is an important aspect for complex analysis with complex coefficients. 

Now consider this equation: f (x) = e(1-x2) in which f (0) = 0. This equation can infinitely differentiate anywhere when it is considered as a real function. But when it is considered as a complex function it cannot be differentiated at x = 0. Laurent Series can be obtained by replacing x with -1/x2 for the exponential function in that power series. Notably, the Laurent series can be utilized for the expression of a holomorphic function defined on an annulus. On the other hand, power series can be utilized for the expression of holomorphic functions defined on a disc.  Now consider an equation with a complex coefficient such as:

n=-∞∑∞an (z-c)n

  Here an is the complex coefficient and c is the complex center. 

Taylor series

Brook Taylor developed the Taylor Series in 1715. Taylor Series refers to an infinite summation of a function and this summation introduces the derivatives of the function in a single point. Taylor Series also can be called as Maclaurin Series. In the 18th century, Colin Maclaurin extended the use of special cases of the Taylor Series when derivatives have been considered at the 0 points. Taylor series can be developed as:

“f (a) + f’ (a)/1! (x-a) + f” (a)/2! (x-a)2 + f’’’ (a)/3! (x-a)3 + ………….”

In this series the “n!” is denoted the factorial of n. Here n = 1, 2, 3, 4 ……

According to the sigma notation this series can be expressed as:

n=0∑∞fn (a) / n! (x-a)n

Here fn (a) is denoted as the nth derivative of the function f (a). The order zero derivative of this function f (a) is expressed as f itself. It also should be mentioned that “0!” and (x-a)0 can be written as 1. This series can be called the Maclaurin Series when a = 0.

The summation of the first n+1 terms of a Taylor Series is called the nth Taylor Polynomial of that function.  

Laurent series examples

Consider the following equation: f(z) = 1/[(z-1)(z-2i)] = {(1+2i)/5}[1/(z-1) – 1(z-2i)]

At the values of z as 1 and 2i this equation has singularities where the denominator of this function is 0. 

Taylor series examples

The Taylor Series for 1/(1-x) is the geometric series

1+x+x2+x3+x4+ ……

Here for 1/x at a = 1 this series can be written as:

1 – (x-1) + (x-1)2 – (x-1)3 + …..

The Taylor Series can be expressed for ln x at a=1 as:

(x-1) – ½ (x-1)2 + ⅓ (x-1)3 + ………

The Maclaurin Series for the exponential function ex can be developed as:

“n=0∑∞ xn/ n! = x0/ 0! + x1/ 1! + x2/ 2! + x3/ 3! + x4/ 4! + x5/ 5!+……..”

Conclusion

It can be concluded that Laurent Series and Taylor series have a great impact on the complex functions as well as the analytic functions in mathematics. It also can be concluded that the Laurent series can be expressed as a power series with several negative terms whereas the Taylor Series does not contain any negative terms. In an analysis of a complex function with a power series that contains both the negative terms and positive terms then this power series can be called a Taylor Series. Laurent series can be applied in a complex function when the Taylor Series cannot be applied.