Question

Difference between taylor series and laurent series

Answer 1

Taylor Series

Taylor series is a polynomial equivalent expression for a function which is derivable only about those point for a complex functions at which the function is analytic.

For a function f(z) its Taylor series expansion about a point 'a' is:

$\\ f(z) = \sum _{n = 0} ^{ \infty } \limits \frac{ {f}^{(n)} (a)}{n!} {(z - a)}^{n}$

where, a is a complex number

Laurent Series

Laurent Series is an extension to Taylor series and is derivable about a point even if the function is not analytic at it due to its definition in an annulus rather than a circle where its analytic.

For a function f(z) its Laurent series about a point 'a' is:

$\\ f(z) = \sum _{n = - \infty } ^{ \infty } \limits \frac{ {(z - a) }^{n} }{2\pi i} \oint _{ c} \frac{f(z_{0} )}{( z_{0} - a) ^{n + 1} } d z_{0}$

where, a is a complex number

and, c is a contour taken anticlockwise along any simple curve in the annulus.