Name of function

Question

I am sorry if this is a stupid question, but I am struggling to give the proper name to the following function:

$$\ f(r) = \exp(f_1+f_2r+f_3r^2+f_4r^3+f_5r^4+f_6r^5)$$

I ask as it will be in a presentation I am giving next week!

Thanks

Answer 1

You have plenty of choices. For example, you could say

$$F(r) = \exp(f_1r^0 + \cdots +f_6r^5)$$

and name the function $F(r)$. Likewise you could name it $g(r)$ or $\Xi(r)$ or $\mathfrak{A}(r)$. It generally doesn't matter what name you use for functions, but it's usually best to go for simple things, (like $g$ rather than $\mathfrak{A}$.) Sometimes functions of this type (with exponentiation in conjunction with multipilcation and addition of a variable) are called exponential polynomials.

Answer 2

I don't know that it has any particular name, unless one counts describing it as a composition of a polynomial function and an exponential function.

Certainly it is an entire function, since it's a composition of two entire functions.

Just for fun, let's try applying this to it: \begin{align} & \exp(f_1+f_2r+f_3r^2+f_4r^3+f_5r^4+f_6r^5) \\[10pt] ={} & e^{f_1} \left( 1 + \frac{f_2}{1} r + \frac{f_3 + f_2^2}{2} r^2 + \frac{f_4+3f_3 f_2 + f_2^3}{6} r^3 \right. \\[10pt] & \left. {}\qquad\qquad + \frac{f_5 + 4f_4 f_2 + 3f_3^2 + 6f_3 f_2^2 + f_2^4}{24} r^4 + \cdots\cdots \right) \end{align} For example the coefficient of $r^4$ comes from the fact the numbers of partitions of a set of four members corresponding to each of the several partitions of the integer $4$: $$ \begin{array}{rl} 1\text{ partition of the form} & 4 \\ 4\text{ partitions of the form} & 3+1 \\ 3\text{ partitions of the form} & 2+2 \\ 6\text{ partitions of the form} & 2+1+1 \\ 1\text{ partition of the form} & 1+1+1+1 \end{array} $$

It is a bit confusing to use indices that don't match the powers of $r$; i.e. $\exp(f_0+f_1 r + f_2r^2+f_3r^3+f_4r^4+f_5r^5+f_6r^6)$.

Answer 3

Just call it the exponential of a polynomial. Most functions aren't common enough to get their own names.