Power series of exp
In the real case it is relatively easy to visualize how a certain function is approximated by its Taylor series expansion in the neighbourhood of a given point.
Now we will show the same phenomena in the complex case. The partial sums of the Taylor series (Taylor polynomials) centered at zero are shown step-by-step on the picture below in the case of the exponential function.
The series and polynomials at hand have the following form: \[ \exp\,z = \sum\limits_{k=0}^{\infty} \frac{1}{k!} z^k, \] \[ T_n\exp\, (z) = \sum\limits_{k=0}^{n} \frac{1}{k!} z^k = 1 + z + \frac{1}{2} z^2 + \frac{1}{3!} z^3 + \dots + \frac{1}{n!} z^n. \] After loading the page, the polynomial corresponding to n=0, i.e. the constant 1 can be seen. (Please click on the links More or Less!)
\(f(z) = T_{n}\exp\,(z),\quad z \in \mathbb{C}(0,8)\)
Compare the Taylor polynomials with the picture of the exponential function plotted on the same domain. (The coloring is also shown.)
\(f(z) = \exp\,z,\quad z \in \mathbb{C}(0,8)\)
One may see how the polynomials of higher degree give a fine approximation near zero.