Taylor series

$$f\left(x\right)=\sum _{k=0}^{\infty }{c}_k{x}^k=c_0+c_{1}x+c_2{x}^2+\cdots .$$

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

$f(x\pm h)=f\left(x\right)\pm h{f}^{\prime }\left(x\right)+\frac{h^2}{2}{f}^{\prime \prime }\left(x\right)\pm \frac{h^3}{6}{f}^{\prime \prime \prime }\left(x\right)+O\left(h^4\right)$

Let’s think about $\sin \left(x\right)$.

$$f\left(x\right)=\sin \left(x\right)f\left(0\right)=0,f^{\prime }\left(x\right)=\cos \left(x\right)f^{\prime }\left(0\right)=1,f^{\prime \prime }\left(x\right)=-\sin \left(x\right)f^{\prime \prime }\left(0\right)=0$$

Thus,

$$\begin{array}{ccc}\sin \left(x\right)& =0+\frac{1}{1!}x+\frac{0}{2!}{x}^2+\frac{-1}{3!}{x}^3+\cdots & =x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots ,\end{array}$$

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

https://betterexplained.com/articles/taylor-series/

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

$$\exp \left(z\right)=e^z,z=a+bi$$

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 $\exp \left(x\right)\exp \left(y\right)=\exp (x+y)$. 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 $\exp \left(x\right)={\Sigma }_{n=0}^{\infty }\frac{Z^n}{n!}$ 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.