Exponential function in the context of Thermionic diode

In mathematics, exponentiation, denoted b, is an operation involving two numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b is the product of multiplying n bases:$${\displaystyle b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.}$$In particular, ${\displaystyle b^{1}=b}$.

The exponent is usually shown as a superscript to the right of the base as b or in computer code as b^n. This binary operation is often read as "b to the power n"; it may also be referred to as "b raised to the nth power", "the nth power of b", or, most briefly, "b to the n".

Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.

In ecology, the species discovery curve (also known as a species accumulation curve or collector's curve) is a graph recording the cumulative number of species of living things recorded in a particular environment as a function of the cumulative effort expended searching for them (usually measured in person-hours). It is related to, but not identical with, the species-area curve.

The species discovery curve will necessarily be increasing, and will normally be negatively accelerated (that is, its rate of increase will slow down). Plotting the curve gives a way of estimating the number of additional species that will be discovered with further effort. This is usually done by fitting some kind of functional form to the curve, either by eye or by using non-linear regression techniques. Commonly used functional forms include the logarithmic function and the negative exponential function. The advantage of the negative exponential function is that it tends to an asymptote which equals the number of species that would be discovered if infinite effort is expended. However, some theoretical approaches imply that the logarithmic curve may be more appropriate, implying that though species discovery will slow down with increasing effort, it will never entirely cease, so there is no asymptote, and if infinite effort was expended, an infinite number of species would be discovered. An example in which one would not expect the function to asymptote is in the study of genetic sequences where new mutations and sequencing errors may lead to infinite variants.

A diode is a two-terminal electronic component that conducts electric current primarily in one direction (asymmetric conductance). It has low (ideally zero) resistance in one direction and high (ideally infinite) resistance in the other.

A semiconductor diode, the most commonly used type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals. It has an exponential current–voltage characteristic. Semiconductor diodes were the first semiconductor electronic devices. The discovery of asymmetric electrical conduction across the contact between a crystalline mineral and a metal was made by German physicist Ferdinand Braun in 1874. Today, most diodes are made of silicon, but other semiconducting materials such as gallium arsenide and germanium are also used.

The number e is a mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted ${\displaystyle \gamma }$. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.

The number e is of great importance in mathematics, alongside 0, 1, π, and i. All five appear in one formulation of Euler's identity ${\displaystyle e^{i\pi }+1=0}$ and play important and recurring roles across mathematics. Like the constant π, e is irrational, meaning that it cannot be represented as a ratio of integers. Moreover, it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. To 30 decimal places, the value of e is:

In mathematics, an expression or formula (including equations and inequalities) is in closed form if it is formed with constants, variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions.

In mathematics, the factorial of a non-negative integer ${\displaystyle n}$, denoted by ${\displaystyle n!}$, is the product of all positive integers less than or equal to ${\displaystyle n}$. The factorial of ${\displaystyle n}$ also equals the product of ${\displaystyle n}$ with the next smaller factorial:$${\displaystyle {\begin{aligned}n!&=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\\&=n\times (n-1)!\\\end{aligned}}}$$For example,$${\displaystyle 5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.}$$The value of 0! is 1, according to the convention for an empty product.

Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of ${\displaystyle n}$ distinct objects: there are ${\displaystyle n!}$. In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science.

In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean, which uses their sum). The geometric mean of ⁠${\displaystyle n}$⁠ numbers is the nth root of their product, i.e., for a collection of numbers a₁, a₂, ..., a_n, the geometric mean is defined as

When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm ⁠${\displaystyle \ln }$⁠ of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using the exponential function ⁠${\displaystyle \exp }$⁠,

In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various formulas and is also given by the alternating harmonic series. The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) truncated at 30 decimal places is given by:

The logarithm of 2 in other bases is obtained with the formula

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup ${\displaystyle \cup }$ (or a straight line like a linear function), while a concave function's graph is shaped like a cap ${\displaystyle \cap }$.

A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function ${\displaystyle f(x)=cx}$ (where ${\displaystyle c}$ is a real number), a quadratic function ${\displaystyle cx^{2}}$ (${\displaystyle c}$ as a nonnegative real number) and an exponential function ${\displaystyle ce^{x}}$ (${\displaystyle c}$ as a nonnegative real number).