Rational function

A rational function is a function in mathematics, the term in the form

N with natural numbers and write z can, so as the quotient of two polynomials can be represented. So the function is a quotient of two quite rational functions. The numbers can be any real numbers (or complex numbers); the only restriction is that it must be. The rational functions are among the meromorphic functions.

6.1 The field of rational functions on an algebraic variety

Classification

If the denominator polynomial of degree, so constant, so we speak of a rational integral function or by a polynomial function.
Is it possible to represent the function term only with a denominator polynomial of degree, it is a rational function. If and so it is a real rational function.
If and so it is a spurious rational function. It may be spread over polynomial division in a very rational function and a real rational function ( see below).

Examples of rational functions with different degrees counter z and denominator degrees of n:

Asymptotic behavior

For the behavior of infinity or the degrees of the numerator and denominator - polynomial are crucial:

For it

Against if, where is the sign function.
To if ( the asymptote is parallel to the axis),
Against ( the axis is horizontal asymptote ) if,

For results in the second and third cases, each the same limit as for. In the first case one has to consider the numerator and denominator in more detail:

If straight, the result is the same limit as for.
Is odd, so changes in comparison with the sign of the limit.

As described below, you can in the first case the function term by means of polynomial always decomposed into a polynomial and a real fractional rational term; the polynomial then describes a so-called Asymptotenkurve. ( The behavior of the function values can then also get easier, by examining only the behavior of Asymptotenkurve. ) In the special case, an oblique asymptote results.

Examples:

In the fractional rational function is the numerator and the denominator, the limit for being so.
The rational function has the numerator and the denominator degrees; there and here, the result for the equation of the horizontal asymptote.
The rational function has the numerator and the denominator; with the coefficients and thus results: for. Because here is odd, it follows for the limit of the reversed sign, ie. This function can also be written as, that is, the ( oblique ) asymptote has the equation ( and it also follows easily again the limiting behavior just described ).

Curve Sketching

Based in expressions of the rational function following information is true for function graphs make (curve discussion).

Symmetry

A polynomial function (all rational function ) is even / odd if all exponents / are just odd. Are numerator and denominator of one of these two types, as well as the rational function is even or odd:

And are both even or both odd, is straight ( that is, the graph is symmetrical about the y-axis)
If even and odd, so is odd (ie, the graph is point symmetric about the origin ); the same applies if is odd and is even.

In all other cases, ie if counter or denominator function, or both, neither are still just odd symmetry properties are to be decided by more difficult. (See curve sketching and symmetry in geometry ).

Examples:

The graph function is symmetric about the origin, as odd and even, the function is a total odd.

The graph of the function is symmetrical about the y-axis, and since both odd, the total function is therefore straight. This can also be seen differently: When excluding in the numerator and denominator of each x, the function term can be shortened to; now are and straight, the function ie a total turn straight.

When you graph the function with the first term is no symmetry recognizable ( is odd, but not odd or even ); it may show, however, that the graph symmetrical to the point P ( 1 | 1); it is namely: and,

Definition range, zeroes and poles

The rational function is not defined at the zeros of the denominator function.

The zeros of a fractional rational function are determined by those zeros of the numerator function that belong to the domain of the entire function.

A special case arises when a real number is zero of the numerator and the denominator polynomial time. Then the numerator and denominator are by the factor associated linear ( or even more times) divided, i.e. the term can function with this factor (possibly multiple) are reduced.

Come more often than before in the denominator in the numerator, so a pole is present (the corresponding power is then called the multiplicity of the pole );
Otherwise the rational function on the site has an ever -recoverable definition gap, and you can continue the function steadily

Examples:

The function has the domain of definition, since the denominator function has the zero point and the zero point, since that is the only zero of the counter function ( and heard ). is a (double) pole.

The function has the domain of definition. Here, however, is now a zero of the numerator and the denominator function. To shorten the corresponding linear factor, we factored the numerator and denominator first ( by factoring out and applying the binomial ); which leads to or after to cut back. This results in: is a (simple ) pole, however, a steady recoverable definition gap of, and has the zero point (note: is not a zero of, since this value is not to be heard ). For the continuous extension of results: and.

Asymptote

The polynomial division of by using polynomials and are obtained, the degree of less than that of. The asymptotic behavior of

Is thus by the very rational function ( " Asymptotenfunktion " ) is determined ( the actual implementation of the polynomial is only at 3 and 4 required):

Derivation

To derive fractional rational functions you must use the quotient rule in general; In addition, often the chain rule to be useful, for example, when the denominator is a power function, a binomial. Before deriving it is often advisable to first rewrite the function term with the help of a polynomial division and cut the remaining real fractional rational term.

Examples:

During the function, it makes sense, in addition to the quotient rule to apply the chain rule, first place in the denominator apply the first binomial formula. Using the chain rule results in first for the derivative of the denominator function ( in the quotient rule, usually referred to ): ,

The function term is brought by means of a polynomial first on the form,

Antiderivative

In contrast to the quite rational functions with fractional rational functions, it is often quite difficult to find an antiderivative. For this you can, depending on the shape of the fractional rational function, inter alia, apply the following rules (usually you have to the function term by transformations and / or substitution at first into a form suitable to bring ):

Often can be helpful in determining a parent function, the partial fraction expansion.

Examples:

Wanted to be a primitive function. By means of a polynomial can be the first to describe: .

Wanted to be a primitive, with -0.5 to 0.5 should lie. Again, we first rewrite the function term by means of a polynomial: .

Wanted to be a primitive function. This can also be written as with.

A stem function can be defined by means of the substitution, after having transformed the denominator using quadratic addition:
A stem function can be obtained using the partial fraction decomposition, after having initially factored the denominator:

Applications

Rational functions have many applications in science and technology:

Many sizes are inversely proportional to each other, one of the sizes is thus a rational function of the other, wherein the counter and the denominator is a constant (homogeneous ) linear function. A few examples: The velocity v and for a fixed distance s time required t are inversely proportional to each other:
The concentration c of the substance is at a fixed amount of substance n is inversely proportional to the volume V of the solvent:
Acceleration and mass are inversely proportional to each other at a fixed force F: .
For the capacitance C of a parallel plate capacitor is considered as a function of plate spacing d: with the area A of the plates, the electric field constant and the permittivity.

By means of the equation of the optical lens can be represented as a function of the focal length f of the object distance and image distance g b :; Switch to g or b provide a very similar function, but with - instead of .
For the total resistance R of a parallel connection of two resistors and the result is:; an analogous formula is valid for the series connection of two capacitors.
In mechanics arises when one hangs two springs with spring constants and each other for the entire spring constant D of the arrangement:

Different meaning in abstract algebra

In abstract algebra, the concept of a rational function in a more general and somewhat different meaning is used. And that is meant by a rational function in n variables over a field K is an element of the quotient field of the polynomial ring. This quotient field is called Rational Function Fields. In general, a rational function is not a function of some sort, but a ( formal ) fraction of two polynomials. The difference is, however, only over finite noticeable: for example, for each prime p over the finite field ( the body of all residual classes of integers modulo p ) of the fracture a well-defined rational function in the variable X, but no function in the proper sense of the term, because you may use this function in a single value, without the denominator is 0. ( For it is any one in this " " feature, you get what is undefined because the denominator after Fermat's little theorem is equal to 0. ) Always over infinite bodies, however, is a rational function is a function that can have a definition gaps while, but this definition gap is very small compared to the domain of definition. This idea is formalized with the notion of Zariski topology: the definition gap is a Zariski - closed set, and the closed hull of the domain is the whole lot ).

The field of rational functions on an algebraic variety

Be an algebraic variety defined by polynomials, ie

The ring of entire functions is. The field of rational functions is the quotient field of the ring of entire functions.

Exponentiation Parameter (disambiguation) Voltage divider Electric power Cuboid

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