To design some type of mathematical model, I need a set of functions that diverge within a finite range but its integral is a real number. More specifically, what I need is a design of the form:
My model requires two functions of the form $f(x)$ and $\frac{1}{f(x)}$, one of these functions connects from (0,1) to (1,0). Also, as adjust the parameters in the family of functions, functions need to be able to express all the slopes in the y-intercepts, like above two functions.
I found that the following types of functions satisfy this condition.
$f(x)=(1-f(x))^{-\frac{1}{k}}$
where $f(x)$ is a polynomial with positive finite integer degree $m = g(x)$, which is equal or smaller than $k$. For example, setting $f(x) = x^k$ gives the exact same condition I want. In this case, parameter must be $k$ and we can obtain any y- intersect slopes by changing $k$.
However, this type of function cannot be used in my computation environment where very few computational resources are allowed, since the value of the integral including the divergent region is usually expressed as a gamma function.(I tried to compute the gamma function with as few resources as possible, but it was not available.)
So my problem is that : I want to find a function $f(x)$ that satisfied
- $f(x)$ is function that $[0, 1] \rightarrow [1, \infty)$ with $f(0) = 1$ and $\lim_{x \rightarrow 1} f(x) = \infty$, so $\frac{1}{f(x)}$ is function that $[0, 1] \rightarrow [0, 1]$ with $f(0) = 1$ and $f(1) = 0$.
- As change the parameter k, the slopes at the y-intercept of $f(x)$ and $\frac{1}{f(x)}$ must be able to represent the entire range of positive and negative numbers, respectively.
- $\int_{0}^{1}{f(x)dx}$ is finite and easy-to-calculate. More specifically, it should be easy to find from the integral the parameter $k$ that determine which function the objective function is in the function set.
- $f(x)$ and $\frac{1}{f(x)}$ must be smooth in their domain.
My mathematical background is shallow and I can't think of various functions, so I want to get some advice. Any type of function is fine as long as it is easy to compute. Thank you.