The invention of the "$h$" function is related to my previous three works. The first work is my original quest for indeifnite integration of the Gaussian integral. I applied methods of saddle-point approximation, generating function, ordinary differential equation, and combinatorics to tackle this problem.
The second work is to solve an integral problem related to "general Gamma function" (GGF). The reason to focus on this integral is because I found that the Gassuian integral can be reduced to a GGF problem. This work is primarily a combinatoric study.
The third work is a short piece to explain how the "$h$ " function can solve indefinite integration of the Gamma, Gaussian, and Exponential Integrals. In this piece, I conducted a historical study to explain why previous studies all failed to recognize the "$h$" function.
Given the fact that I make no effort to earn academic credit, you can simply ignore my works if you believe I am plainly wrong. On the other hand, I am very happy to discuss the detail of these works to whoever is interested and also welcome any feedback. My only purpose is to improve these works and spread out the idea if my claim is tenable.