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  2. Exponential growth - Wikipedia

    Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time ...

  3. Euler's formula - Wikipedia's_formula

    The exponential function e x for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of e z for complex values of z simply by substituting z in place of x and using the complex algebraic operations.

  4. Exponential decay - Wikipedia

    Exponential decay is a scalar multiple of the exponential distribution (i.e. the individual lifetime of each object is exponentially distributed), which has a well-known expected value. We can compute it here using integration by parts .

  5. e (mathematical constant) - Wikipedia

    If f(t) is an exponential function, then the quantity = / ′ is a constant, sometimes called the time constant (it is the reciprocal of the exponential growth constant or decay constant). The time constant is the time it takes for the exponential function to increase by a factor of e : f ( t + τ ) = e f ( t ) {\displaystyle f(t+\tau )=ef(t)} .

  6. Biological exponential growth - Wikipedia

    Biological exponential growth. Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. Most commonly apparent in species that reproduce quickly and asexually, like bacteria, exponential growth is intuitive from the fact that each organism can divide and ...

  7. Euler's identity - Wikipedia's_identity

    In mathematics, Euler's identity [note 1] (also known as Euler's equation) is the equality. where. e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i2 = −1, and. π is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss ...

  8. Double exponential function - Wikipedia

    A double exponential function (red curve) compared to a single exponential function (blue curve). A double exponential function is a constant raised to the power of an exponential function. The general formula is (where a >1 and b >1), which grows much more quickly than an exponential function. For example, if a = b = 10:

  9. Exponential function - Wikipedia

    Exponential functions with bases 2 and 1/2. The exponential function is a mathematical function denoted by () = ⁡ or (where the argument x is written as an exponent).Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras.