exponential function — from formula anatomyThe exponential function, often written as exp(x) or e^x, is a mathematical function where the rate of growth is proportional to the current value. It is defined as the unique function that is its own derivative, with the base e (approximately 2.71828). This function appears throughout physics whenever a quantity changes at a rate proportional to itself, such as radioactive decay, population growth, or the cooling of a hot object.
The exponential function was introduced by Leonhard Euler in the 18th century, who also established the relationship e^(iπ) + 1 = 0, linking it to trigonometry and complex numbers.
The exponential function can be defined by the infinite series e^x = 1 + x + x^2/2! + x^3/3! + ..., which converges for all real and complex x. Its defining property, d/dx e^x = e^x, makes it the solution to the differential equation dy/dx = ky, where k is a constant. This equation models processes where the change rate is proportional to the amount present, leading to exponential growth (k>0) or decay (k<0).
Exponential functions are used in physics to describe radioactive decay, capacitor discharge, atmospheric pressure with altitude, and the time evolution of quantum states in the form of complex exponentials.
The exponential function is so important that it appears in the Schrödinger equation, where the time evolution of a quantum state is given by a complex exponential of the Hamiltonian.
| f(x_i) | variable | model prediction at input x_i — function value |
| y_i | variable | observed data point — data value |
| σ_i | variable | standard deviation of measurement error — standard deviation |
| ∏ | operator | Product over all data points i — product operator |
| exp | operator | Exponential function — exponential function |
| N_i | variable | Number of particles in state i — particle number |
| N | variable | Total number of particles — particle number |
| g_i | variable | Degeneracy of state i — degeneracy |
| E_i | variable | Energy of state i — energy joule |
| T | variable | Absolute temperature — temperature kelvin |
| k | constant | Relates energy to temperature at the particle level — boltzmann constant = 1.380649e-23 joule per kelvin |
| e^{-E_i/kT} | operator | Exponential of the negative energy over kT — exponential function |
| x | variable | input point — input variable |
| x' | variable | another input point — input variable |
| σ | variable | signal amplitude — amplitude |
| ℓ | variable | length scale — length scale |
| exp | operator | raises e to the given power — exponential function |
| s | variable | sigmoid function value — sigmoid function |
| ζ | variable | input argument — argument |
| a | variable | steepness parameter — steepness parameter |
| exp | operator | raises e to the given power — exponential function |
| P(v) | variable | Probability density of speed v — probability density second per metre |
| v | variable | Speed of a particle — speed metre per second |
| σ | variable | Scale parameter related to temperature — speed scale metre per second |
| exp | operator | Exponential function — exponential function |
| √ | operator | Square root — square root |