and Related Statistical Applications

### We don't need "statistical tables" to conduct significant tests.

This might be trivial in terms of the outcome given the fact that we can easily conduct a significant test with a computer. But this matters a lot when we teach elementary statistics to college students. For many undergraduates, they don't know why we can not directly compute a p-value manually with a formula. If they know calculus well, they might know from textbooks that the formula for the cumulative distribution function of the Z (standard normal) distribution is not existed. However, my research does propose a formula

\begin{equation*}

\Phi \left( z \right)=\frac{1}{2}\left[ 1+\frac{z}{\sqrt{2\pi
}}{{e}^{\tfrac{-{{z}^{2}}}{2}}}h_{\tfrac{-1}{2}}^{\tfrac{-{{z}^{2}}}{2}} \right].

\end{equation*}

For instance, when we conduct a Z- test with $z=2$, we can derive $p(z\le 2)=0.9772$ by using a Z-table (from wikipedia: http://en.wikipedia.org/wiki/Standard_normal_table). However, we can also use the above formula to derive the same result.

\begin{align*}

p(z\le 2)&=\frac{1}{2}\left[ 1+\frac{z}{\sqrt{2\pi }}{{e}^{\tfrac{-{{z}^{2}}}{2}}}h_{\tfrac{-1}{2}}^{\tfrac{-{{z}^{2}}}{2}}
\right] \\

& =\frac{1}{2}\left[ 1+\frac{2}{\sqrt{2\pi }}{{e}^{-2}}h_{\tfrac{-1}{2}}^{-2} \right] \\

& =\frac{1}{2}\left[ 1+\frac{2}{\sqrt{2\pi }}\cdot \left( \text{0}\text{.135335283236613} \right)\cdot \left(
\text{8}\text{.839439240919045} \right) \right] \\

& =\text{0}\text{.977249868051822}

\end{align*}

So, if our calculator has built a button "$h$" (similar to exponential number "$e$"), we can simply compute a Z-probability with the above formula.

### The definition of the elementary function is too narrow.

The conventional wisdom among professional mathematicians is that the elementary function only contains " polynomial function", "trigonometric function", "exponential function", and "logarithmic function". Therefore, many integrals such as the gamma integral, exponential integral, and Gaussian integral do not have a closed-form anti-derivative. I believe this theorem is contingent on the definition of "elementary function" and the current definition is somewhat arbitrary. In fact, the "$h$" function is also a convergent infinite series similar to trigonometric, exponential, or logarithmic function. My research demonstrates that if we can accept the "$h$" function as another elementary function, then many of these unsolvable integrals do have a closed-form antiderivative.

### Unifying unsolvable integrals

The "$h$" function can serves as the minimal denominator for most of the commonly-used distributions such as the gamma function, the exponential integral function, the error function, the beta function, the hypergeometric function, the Marcum Q-function, and the truncated normal distribution.

### Mathematical clarity

I also discover the basic algebraic rules for the "$h$" function. This allows us to directly work with these "unsolvable" integrals in mathematical deductions and increases mathematical clarity. For example, I have demonstrated in (3.14) that the derivative and integral operators are reversible for the gamma integral.