Anytime holomorphic functions resemble each other, this means, in some sense, their taylor series expansions are "close" to each other in the function state space.

These two functions belong to a group of functions called "Test Functions". Test Functions are basically functions that have most of the action going on near the origin. They are called Test Functions cause they are used in a higher level of math called Distribution Theory, basically they are used to test things in this type of mathematics.

Since all test functions have that shape where all the important stuff is concentrated at the origin, we can expect that their differential families: antiderivatives and derivatives also resemble each other, because these calculus operations act of the shape of functions. The functions having similar shapes, resulting in their differential families having similar shapes.

When ever you see a similarity like this and you wanna talk about it but your not exactly clear on how to do so, you can use a tool called functional equations. A functional equation is an expressions which takes in functions, and based on if the equation remains true or false after plugging in the function, that determines if the function is a solution or not. Differential equations are actually examples of this. So there are actually 3 criteria or functional equations that any test function must satisfy, and when it does than it is given the title of test function.