If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces to

Explanation:

Taylor Series:

If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-

 where 

Laurent Series:

If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C₁ and C₂ having centre at 'a' & respective radii r₁ and r₂ (r₁ > r₂).

The negative part of Laurent's series i.e  is called the singular part, and if that vanishes the terms that remain will be  , which is nothing but Taylor series.

Important Points

Maclaurin's Series:

When a = 0 in Taylor's series i.e. about origin or powers of z, the series formed is Maclaurin's series.

Cauchy's Theorem:

If f(z) is single-valued and an analytic function of z and f'(z) is continuous at each point within and on the closed curve c, then according to the theorem, .