# Why does Euler's Number appear in so many equations? [closed]

Is there an intuitive explanation for $$e$$ in those randomly chosen equations from my theoretical chemistry book:

$$\Phi(x,t)=(2\pi \hbar)^{-1/2}\int_{-\infty}^\infty C(p){\color{red}e}^{i(px-E(p)t)/\hbar} \, \mathrm{d}p$$ $$\psi(x,t) = \phi(x){\color{red}e}^{-iEt/\hbar}$$ $$u_{ij}\equiv u_{ij|z}=(\overline{e}/2)|\mathcal E|\int \Phi_i^\star \cdot z \cdot {\color{red}e}^{ikx}\cdot \Phi_j \, \mathrm{d}V$$

I am aware that Euler's number is the base of an exponential function whose rate of growth is always proportionate to its present value, but it always bothers me that I can't really tell how it finds its way into such equations.

• What makes you think it is more suited for the chemistry SE site and not the mathematical one? Commented Aug 7, 2023 at 17:20
• Chemists seldom care about the Euler number, they have rather applied approach to exp and ln functions. Commented Aug 7, 2023 at 17:31
• Well, the Euler number is in none of those equations explicitly. And for using of the respective functions, you need not to know it either. Commented Aug 7, 2023 at 18:04
• Hello! This is a question that exceeds chemistry and science. However, you partly answered it. $\exp(x)$ is a particular solution to the differential equation $\mathrm{d}y/\mathrm{d}x = y$, so Euler's number appears. The second equation of your image is when you apply separation of variables to the time-dependent Schrodinger's equation (a PDE). The differential equation with respect to time, has the mentiond form. Commented Aug 7, 2023 at 18:26
• Thus, from an epistemological point of view, the question may be formulated: if you consider that reality $=$ our conceptualization, why so many chemical and physical processes follow this simple differential equation? Other examples are first-order irreversible kinetics, radioactive decay, and the list goes on... Commented Aug 7, 2023 at 18:27

Euler's number $$e$$ as such is less prominent than the exponential function $$f(x)=\exp(x)$$. It is this function, not $$e$$, that appears widely. For instance, the exponential function appears ubiquitously in assessing the asymptotic growth or decay of functions or populations.
In the real domain we certainly have $$\exp(x)=e^x$$, which is a consequence of the familar limit for $$e$$ combined with the algebraic properties of the exponential function. But that equality does not carry over into complex analysis. The function $$\exp(z)$$ is defined as a single-valued Taylor series that converges for all $$z$$, or equivalently as the unique solution to $$df/dz=f, f(0)=1$$; whereas $$e^z$$ is a branched function just like $$a^z$$ for other constants $$a$$. Mathematicians who deal with complex variables are well aware of the dustinction between the exponential function and $$e$$.