# Taylor series

$f(x) = \sum_{k=0}^\infty c_k x^k = c_0 + c_1 x + c_2 x^2 + \dotsb.$

This is an approximation that is a function of h and derivatives of $$f(x)$$ are elements of parameters.

$$f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f'''(x) + O(h^4)$$

Let’s think about $$\sin(x)$$.

$f(x) = \sin(x) \ f(0) = 0, f'(x)=\cos(x)\ f'(0)=1, f''(x)=-\sin(x)\ f''(0)=0$

Thus,

\begin{align*} \sin(x) &= 0 + \frac{1}{1!}x + \frac{0}{2!}x^2 + \frac{-1}{3!}x^3 + \dotsb &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dotsb, \end{align*}

This is approximation. Now $$x$$ becomes $$h$$ and parameters calculated from derivatives of $$f(x)$$ at $$0$$.
$$f(x \pm h) = f(x) \pm hf'(x) + \frac{h^2}{2}f''(x) \pm \frac{h^3}{6}f'''(x) + O(h^4)$$

Taylor series and Newton’s bionomial theorem explain the complex exponent.

$\exp(z) = e^{z}, \ z = a+bi$

The imaginary exponent is hard to understand intuitively. The exponential function $$e^{x}$$ on a complex domain can be regarded as a function exp(x) that behaves like exponential function, i.e. a product of functions is addion of arguments $$\exp(x) \exp(y) = \exp(x+y)$$. The product of $$\exp$$ fucntion becomes addition of arguments by Newton’s binomical theorem. The costomary expression is $$e^{x}$$. This can be done when $$\exp(x) = \Sigma ^{\infty}_{n=0} \frac {Z^{n}}{n!}$$ The taylor series with repidly decaying pactorial coefficients $$n!$$. This series converges absolutely for every complex $$z$$ and converges uniformly on every bounded subset of the complex plain. Rudin’s Real and complex analysis.

##### Jun Kang
###### Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.