What is order and Type of an entire function? Why type is considered as a more delicate characteristic of entire function?

Question

What is order and Type of an entire function? Why type is considered as a more delicate characteristic of entire function? I know the definition of order and type but can not understand the physical aspect of it.

Answer 1

The order is a measure of the growth at $\infty$. If $f$ is of order $\rho$, its growth at $\infty$ is like $e^{|z|^\rho}$. If $\rho>0$, for any small $\epsilon$ there are constants $C_1,C_2$ such that $$ C_1e^{|z|^\rho-\epsilon}\le|f(z)|\le C_2e^{|z|^\rho+\epsilon}. $$

The type is a subtler measure of growth for functions of the same order. Essentially, $f$ is of order $\rho$ and type $\sigma$ if it grows at $\infty$ like $e^{\sigma|z|^\rho}$. If $\sigma>0$, for any small $\epsilon$ there are constants $K_1,K_2$ such that $$ K_1e^{(\sigma-\epsilon)|z|^\rho\epsilon}\le|f(z)|\le K_2e^{(\sigma+\epsilon)|z|^\rho}. $$

For example, $f(z)=e^{az^n}$, $n\in\Bbb N$, $a>0$ has order $n$ and type $a$.