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 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.
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.
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.