Exponential function

In mathematics, a function is called the exponential function of the form with a real number as a base (base ). In its most common form, the real numbers are allowed for the exponent. In contrast to the exponential functions, where the base is the independent variable ( Variable), wherein the variable of the exponential exponent (also high- speed ) of the power of expression. Referred to the naming. Exponential functions have in the natural sciences, for example in the mathematical description of growth processes, of outstanding importance (see exponential growth ).

As the exponential function in the strict sense (more precisely actually natural exponential ) refers to the exponential function, ie, the exponential function with the Euler's number as a base; commonly used for this is also the spelling. This feature has special properties compared to the other exponential functions. Using the natural logarithm can be calculated using the equation each exponential function in such a lead back to the base. Therefore, this article deals mainly with the exponential function to the base.

• 8.3.1 Estimation down
• 8.3.2 estimate upwards
• 8.3.3 Derivation of the exponential function
• 8.3.4 Growth of the e- function compared to polynomial

• 9.1 inverse function
• 9.2 Differential Equation
• 9.3 Examples 9.3.1 Physics
• 9.3.2 Chemistry
• 9.3.3 Biology
• 9.3.4 Stochastic
• 9.3.5 economic

Definition

The exponential function to the base may be defined on real numbers in different ways.

A possibility is to define as a power series, the so-called exponential

The Faculty of designated.

Another possibility is to define as a limit value of a sequence comprising:

Both types are used in the definition of the complex exponential function to the complex numbers suitable (see below).

The real exponential function is positive, continuous, strictly increasing and surjective.

It is therefore bijective. Therefore its inverse function, the natural logarithm exists.

This also explains the term antilogarithm of the exponential function.

Convergence of the series, continuity

Pointwise convergence of the series used to define the exponential function

Can be shown for all real and complex simple with the ratio test; it even follows absolute convergence. The radius of convergence of the power series is therefore infinite. Since power series are analytic at each interior point of the convergence range, the exponential function is thus in each real and complex point trivially continuous.

Calculation rules

Since the exponential function satisfies the functional equation can be generalized with their help exponentiation on real and complex exponents, by defining:

For any and all real or complex.

In general, this transformation applies to for any other values ​​as a new basis:

Such functions are called exponential functions, and " transform " multiplication into addition. More demonstrate that the following laws:

These laws apply to all positive real and and all real and. Expressions with fractions and roots can often be simplified by using the exponential function:

See also rules for computing logarithms.

Derivation: the importance of " natural" exponential

The importance of e-function, specify the exponential function with base is based on the fact that its derivative is obtained again the exponential function:

Additionally, if you

Calls, the e-function on the real line is even the only function that does this. Thus, one can define with this initial condition, the exponential function as a solution of this differential equation.

More generally follows for real from

And the chain rule, the derivative of any exponential functions:

In this formula can not be replaced with a logarithm to any other base of the natural logarithm; the number e thus goes into the differential calculus on the " natural" way into the game.

Exponential function to the complex numbers

Using the series representation

You can define the exponential function for complex numbers. The series converges absolutely for all.

The exponential reserves for all complex numbers, the following important properties:

The exponential function is thus a surjective, but not injective group homomorphism from the abelian group on the Abelian group, ie of the additive on the multiplicative group of the body.

In the exponential function has an essential singularity, otherwise it is holomorphic, that is, it is an entire function. The complex exponential function is periodic with period 2πi of the complex, it is therefore

Confining their domain on a strip

With, then it has a well-defined inverse function, the complex logarithm.

The exponential function can be used to define the trigonometric functions for complex numbers:

This is equivalent to Euler's formula

The result is derived specifically the equation

The major in physics and engineering complex Exponentialschwingung with the angular frequency and the frequency.

Similarly, the exponential function may be used to define the hyperbolic functions:

One can define a general power also in the complex:

The values ​​of the power function are dependent on the choice of Einblättrigkeitsbereichs of the logarithm, see also Riemann surface. Its ambiguity is indeed caused by the periodicity of its inverse function, just the exponential function. Their basic equation

Arises from the periodicity of the exponential function with a real argument. Their period length is exactly the circumference of the unit circle, the sine and cosine functions describe because of Euler's formula. The exponential, the sine and cosine function are in fact only a part of the same ( generalized to complex numbers ) exponential function, which is obviously not in the real domain.

The exponential function can be generalized to Banach algebras. It is there also on the number

Defined, which converges absolutely for all limited arguments from the Banach algebra under consideration.

The essential characteristic of the real ( and complex ) exponential

Is in this generality, however, only valid for values ​​and which commute, ie for values ​​with ( of course this is always fulfilled in the real or complex numbers, because multiplication is commutative there). Some calculation rules of this kind for the exponentials of linear operators on a Banach space provide the Baker -Campbell - Hausdorff formula.

An important application of this generalized exponential function is found when solving linear systems of differential equations of the form with constant coefficients. In this case, the amount of the Banach matrices with complex entries. Using the Jordan normal form can find a basis or similarity transformation, in which the matrix exponential has a finite computation rule. More specifically, it is a regular matrix, so that, with a diagonal matrix and a matrix nilpotent that commute with one another. It is thus

The exponential of a diagonal matrix, the diagonal matrix of the exponentials, the exponential of the matrix is ​​a nilpotent matrixwertiges polynomial with a degree which is smaller than the dimension of the matrix.

Numerical calculation options

As a fundamental function of analysis was thinking a lot about possibilities for efficient calculation of the exponential function to a desired accuracy. In this case, the calculation is always reduced to the evaluation of the exponential function in a small neighborhood of zero, and work with the beginning of the power series. In the analysis by reducing the necessary working accuracy must be balanced against the number of necessary multiplications of high- precision data.

The rest of the -th partial sum has a simple assessment against the geometric series, which on

The simplest reduction uses the identity, that is, for given is determined, being chosen according to accuracy considerations. This will be, in a certain working precision, calculated and times squared. is now reduced to the desired accuracy and as returned.

More efficient methods presuppose that work better accuracy in addition and ( Arnold Schönhage ) available in any ( occurring on specification). Then the identities

Be used to transform a from the interval or a much smaller interval and thus to reduce the consuming squaring or avoid.

Background and evidence

Motivation

Of the exponential function is encountered when trying to generalize to any real exponent exponentiation. It starts from the computation rule and is therefore looking for a solution of the functional equation with. Referring now first of all that a solution actually exists, and computes its derivative, so we come to the expression

What this mean? If we call this limit, applies to studies

Defined number (or, then that is the logarithm to the base must be ) by the chain rule formally

Then met probably

How can you calculate this number? If, purely formal and solves the equation

Is therefore to be assumed that

Applies.

For obtained with purely formal presentation

Thus, a definition of the exponential function.

Alternatively, you can also try the function

To develop into a Taylor series. Since by induction also

Must apply, ie, one obtains for the Taylor series at the point

Precisely the different definition of the exponential function. Furthermore it is then shown that the so- defined exponential function, actually has the desired properties.

Convergence of the sequence representation

The sequence used to define the exponential function

Is convergent for real pointwise, since it is monotonically increasing and bounded above, first, second, from a certain index.

Proof of the monotonicity

From the inequality of arithmetic and geometric means follows for

The result is therefore monotonically increasing for almost all.

Proof of the boundedness

From the inequality of the harmonic and geometric means follows for

For and the result is therefore limited for all:

For obvious and is the barrier

Functional equation

Here and converge, converge their product

Is now so provides the Bernoulli inequality for sufficiently large

For is obtained from the simple to display and also the inequality for Bernoulli's inequality for sufficiently large

The exponential So actually fulfills the functional equation.

Inequalities

Estimate downward

For real can the exponential function with

Estimate downward. The proof follows from the definition

And the fact that for sufficiently large. Since the sequence is monotone increasing, the limit is therefore strictly greater zero.

This estimate can be the important inequality

Tighten. For it follows from, for example, the proof follows by Bernoulli's inequality to the definition

Applies. An application of this inequality is the Polya - proof of the inequality of the arithmetic and geometric means. However, the inequality of arithmetic and geometric means facilitates the investigation of the episode very much; therefore, to avoid a circular argument, the Polya - proof needed derivations of the exponential function that do the arithmetic and geometric means, without inequality.

Estimate upwards

From the simple to display and inequality for the Bernoulli's inequality, an estimate is obtained for real and sufficiently large upward:

So

Derivative of the exponential function

The most important function of these two estimates, the calculation of the derivative of the exponential function at the position 0:

Together with the functional equation follows from the derivative of the exponential function for arbitrary real numbers:

Growth of the e - function compared to polynomial functions

Often the statement is required that the exponential function grows much stronger than any power function, ie

For this is clear, for either inductively the rule of L'Hospital are used, or also be estimated elegant:

First applies

Because applies

This converges to, and hence the above limit tends to 0

Change of basis

As already mentioned above, applies

Proof: By the definition of the logarithm is equivalent to, from which the identity follows. Supplies by replacing

Wherein the logarithm calculation rule was used for powers in the second step.

The differential equation of the exponential function

To grasp the simple differential equation solve and still requires, so one obtains a definition of.

Inverse function

Does not require one, we used the inverse function of

And because, according to the properties of the logarithmic function

And one can form the inverse function and receives

Since the lower limit is equal to 1, and wherein the inverse function in accordance with property of the inverse function.

Differential equation

If we expand the differential equation and solve for them, we obtain for the form

Is specifically for

Then is a solution and, then

And by assumption

For any we carry

One. The result is

And, by assumption, again

Examples

It now has a powerful tool to describe processes in physics and chemistry, where to get a result, the exponential function containing the mold by means of an approach of the type.

Physics

As examples for the frequent occurrence of the exponential function in the physics can be mentioned:

• Radioactive decay
• Barometric formula
• Temporal charge curves of an electrical capacitor
• Temporal energy curve during the switching of a coil by self-induction

Chemistry

As an example here is outlined in the chemical, a single chemical reaction. It is assumed that we present the solution of a substance, such as cane sugar in water. The cane sugar will now be converted by a catalyst to invert ( hydrolyzed). In this simple chemical reaction, is the rate law (ignoring the reverse reaction ) formulated as follows:

We denote the set of not yet been converted cane sugar, so is the reaction rate, and according to the rate law formulated above, the equation is

With a reaction-specific rate constant. From this moment, the law is obtained according to the above differential equation, an integral law, which provides us with the amount of leftover cane sugar as a function of time:

Where the constant referred to the existing amount of time. The chemical reaction that is asymptotically approaches its final state, the complete conversion of cane sugar into invert sugar. ( The neglect of the back reaction is acceptable here because the chemical equilibrium of sucrose hydrolysis is very strong on the part of invert sugar ).

Biology

• " Exponential " growth of a population of microorganisms, for example,

Stochastics

What is the probability of randomly not to receive one or more coins when coins are randomly distributed on the receiver and is very large?

The Euler number and the approximate formula for the exponential function

Allow a simple estimation.

The probability of obtaining a coin is at the first distribution, and not to receive the coin. The probability twice to get no coin is:. Consequently, the probability times to be ineffective:

The likelihood of only once to succeed, is the product of failures, success and possible combinations when success arrives (the first time, or second or third ...):

The probability of obtaining more than one coin is equal to:

The number of coins, it must be in order to reduce the likelihood not to receive, for example, instead of 0.37 to 0.1? From the above approximate formula follows:

Or in other words: how many coins there must be more than receiver?

So go blank on average only 10% of recipients that 2.3 times the amount of coins is necessary, at 1 %, almost 5 times the number.

Economy

• Continuous compounding
• The denomination is usually followed by an exponential law with the increase of the value. The example of the euro is shown on the points for each coin or bill a best fit line. The slight deviations from these straight lines follow from the requirement for "round" numbers, which shall be specified exactly with only one significant digit ( not to be confused with plain numbers).

Generalizations

If a size is, their potencies for any non - negative integer exist and if the limit exists, it is useful to define the abstract concept of the exponential above. The same applies to operators that, including their powers, a linear mapping of a domain of definition of an abstract space (with elements ) result in a range of real numbers: here it is even useful for all real, all over (more precisely, in the accompanying financial statements range) Exponentialoperatoren to be defined by the expression, where the convergence of this expression initially remains open.