ABSTRACT

In real analysis, many functions f(x) can be expressed as a power series: 2oa, (x-xo)"=a6 + a(x-xo) + as(x-x6) +...These series expansion is called Maclaurin series if xo = 0 and Taylor series in general. For instance ex= I + x + x/2!+ ... Function which can he written as convergent Taylor series are called real analytic. These series are important for a number of reasons, for instance they provide quick ways to compute functions such as ex, providing of course that the series converges. In these work, we explore the theories of power series for complex functions. It turns out that every complex analytic functions can also be written as a convergent Taylor series F (Z)= 2ia, (z-o)" =ao + a, (z-zo) +... but with complex coefficients and rather that real coefficient. Furthermore the· complex analytic functions are only functions with convergent Taylor series expansions. Now suppose f (z) has singularity at z,%. Then f cannot be written as a Taylor series. However, it turns out that if the singularity is .. isolated" f can still be written as a power series if one also a11ows negative powers of zo F(2) = 2oa, (a-s)" =ao + a (z-zo)...These series are known as Laurent's o ,, .•~. ,'.~ ·series. Laurent's series allows one to a better understanding of singularity. ,:- Eventually we be able to classify singularity into several distinct types, such as poles, essentials, removabJe singularities.