On a Review of the Basic Taylor Series Theory with an Application to the Kinetic Equation
Eziokwu, C. Emmanuel *
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Okereke, N. Roseline
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Nwosu Chidinma
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
Ukeje Emenike
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
Given a function f which is analytic on a disk (Z :| Z-Z0 |\(\le\) \(\mathfrak{R}\)) then the power series \(\sum_{n=0}^{\infty} \frac{f^{(n)}\left(z_{0}\right)}{n!}\left(z-z_{0}\right)\) is called the Taylor series of f at z0 and it converges to f on the given disk.
On the above, we did consider those Taylor series that are holomorphic and presented a major result which shows that a power series represents a holomorphic function and the converse of it’s consequence and resultantly observed that for a holomorphic function on G, w\(\epsilon\)G and \(\gamma\)-a positively oriented simple, closed, smooth, G- contractible curve that has w inside \(\gamma\),
\(\int^{(k)}(w)=\frac{k !}{2 \lambda j} \int \frac{f(z)}{(z-w)^{k+1}} d t\)
after which the identity principle and maximum – modulus theorem were considered and finally was the application of the above to integral equation for the homogenous problem and the non linear problem for an analytic function f in fluid mechanics.
Keywords: Taylor series, Cauchy theorem holomorphic functions zeros and identity principle reactor continuity kinetics equation