Exponential Function Properties Derivatives And Differential

As a function of the real variable x, the graph of e^x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x.

Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form ka^x, where a, called the base, is any positive real number. This article will focus initially on the exponential function with base e.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below .

1 Properties

Using the natural logarithm, one can define more general exponential functions. The function

defined for all a > 0, and all real numbers x, is called the exponential function with base a.

Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws:

These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation because:

2 Derivatives and differential equations

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,

That is, e^x is its own derivative, a property unique among real-valued functions of a real variable. Other ways of saying the same thing include:

In fact, many differential equations give rise to exponential functions, including the Schrödinger equationIn physics, the Schrodinger equation proposed by the Austrian physicist Erwin Schrodinger in 1925, describes the time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to New and the Laplace's equationLaplace's equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. Solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics because t as well as the equations for simple harmonic motionSimple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is ch.

Thus any exponential function is a constantIn mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed. Constant number The most widely mentioned sort of constant is a fixed, but possibly unspecified, n multiple of its own derivative.

If a variable's growth or decay rate is proportionalThis article is about proportionality, the mathematical relation. For other uses of the term proportionality see proportionality (disambiguation). In mathematics, two related quantities x and y are called proportional (or directly proportional if there ex to its size — as is the case in unlimited population growth (see Malthusian catastropheA Malthusian catastrophe sometimes known as a Malthusian check is a return to subsistence-level conditions as a result of agricultural (or, in later formulations, economic) production being eventually outstripped by growth in population. Theories of Malth), continuously compounded interestIn finance, interest has three general definitions. Interest is a surcharge on the repayment of debt (borrowed money). Interest is the return derived from an investment. Interest is the right to claim in a corporation such as that of an owner or creditor., or radioactive decayRadioactivity Radioactive decay is the process by which radionuclides decay, emitting ionizing radiation. Such nuclear reactions involve a change in the composition of the nucleus, in contrast to chemical reactions which involve only an exchange or sharin — then the variable can be written as a constant times an exponential function of time.

3 Formal definition

The exponential function e^x can be defined in two equivalent ways, as an infinite series:

or as the limit of a sequence:

In these definitions, stands for the factorial of n, and x can be any real number, complex number, element of a Banach algebra (for example, a square matrix), or member of the field of p-adic numbers.

For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.

4 On the complex plane

When considered as a function defined on the complex plane, the exponential function retains the important properties

for all z and w.

It is a holomorphic function which is periodic with imaginary period and can be written as

where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.

Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:

for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.

5 Matrices and Banach algebras

The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices. In this case we have

e^x is invertible with inverse e^−x

In addition, the derivative of exp at the point x is that linear map which sends u to u · e^x.

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:

where A is a fixed element of the algebra and t is any real number. This function has the important properties

6 On Lie algebras

The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

7 Double exponential function

The term double exponential function can have two meanings:

a function with two exponential terms, with different exponents
a function f(x)=a^a^x; this grows even faster than an exponential function; for example, if a=10: f(-1)=1.26, f(0)=10, f(1)=1e10, f(2)=1e100= googol, f(3)=1e1000, ..., f(100)= googolplex.

Compare the super-exponential function, which grows even faster.

8 See also

exponential growth

Special functions Complex analysis Exponentials

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