Rational function

A function ${\displaystyle f(x)}$ is called a rational function if and only if it can be written in the form

${\displaystyle f(x)={\frac {P(x)}{Q(x)}}}$

where ${\displaystyle P\,}$ and ${\displaystyle Q\,}$ are polynomial functions of ${\displaystyle x\,}$ and ${\displaystyle Q\,}$ is not the zero function. The domain of ${\displaystyle f\,}$ is the set of all values of ${\displaystyle x\,}$ for which the denominator ${\displaystyle Q(x)\,}$ is not zero.

However, if ${\displaystyle \textstyle P}$ and ${\displaystyle \textstyle Q}$ have a non-constant polynomial greatest common divisor ${\displaystyle \textstyle R}$, then setting ${\displaystyle \textstyle P=P_{1}R}$ and ${\displaystyle \textstyle Q=Q_{1}R}$ produces a rational function

${\displaystyle f_{1}(x)={\frac {P_{1}(x)}{Q_{1}(x)}},}$

which may have a larger domain than ${\displaystyle f(x)}$, and is equal to ${\displaystyle f(x)}$ on the domain of ${\displaystyle f(x).}$ It is a common usage to identify ${\displaystyle f(x)}$ and ${\displaystyle f_{1}(x)}$, that is to extend "by continuity" the domain of ${\displaystyle f(x)}$ to that of ${\displaystyle f_{1}(x).}$ Indeed, one can define a rational fraction as an equivalence class of fractions of polynomials, where two fractions ${\displaystyle {\frac {A(x)}{B(x)}}}$ and ${\displaystyle {\frac {C(x)}{D(x)}}}$ are considered equivalent if ${\displaystyle A(x)D(x)=B(x)C(x)}$. In this case ${\displaystyle {\frac {P(x)}{Q(x)}}}$ is equivalent to ${\displaystyle {\frac {P_{1}(x)}{Q_{1}(x)}}}$.

A proper rational function is a rational function in which the degree of ${\displaystyle P(x)}$ is less than the degree of ${\displaystyle Q(x)}$ and both are real polynomials, named by analogy to a proper fraction in ${\displaystyle \mathbb {Q} }$.[1]

Examples of rational functions

Rational function of degree 3, with a graph of degree 3: ${\displaystyle y={\frac {x^{3}-2x}{2(x^{2}-5)}}}$

Rational function of degree 2, with a graph of degree 3: ${\displaystyle y={\frac {x^{2}-3x-2}{x^{2}-4}}}$

The rational function

${\displaystyle f(x)={\frac {x^{3}-2x}{2(x^{2}-5)}}}$

is not defined at

${\displaystyle x^{2}=5\Leftrightarrow x=\pm {\sqrt {5}}.}$

It is asymptotic to ${\displaystyle {\tfrac {x}{2}}}$ as ${\displaystyle x\to \infty .}$

The rational function

${\displaystyle f(x)={\frac {x^{2}+2}{x^{2}+1}}}$

is defined for all real numbers, but not for all complex numbers, since if x were a square root of ${\displaystyle -1}$ (i.e. the imaginary unit or its negative), then formal evaluation would lead to division by zero:

${\displaystyle f(i)={\frac {i^{2}+2}{i^{2}+1}}={\frac {-1+2}{-1+1}}={\frac {1}{0}},}$

which is undefined.

A constant function such as f(x) = π is a rational function since constants are polynomials. The function itself is rational, even though the value of f(x) is irrational for all x.

Every polynomial function ${\displaystyle f(x)=P(x)}$ is a rational function with ${\displaystyle Q(x)=1.}$ A function that cannot be written in this form, such as ${\displaystyle f(x)=\sin(x),}$ is not a rational function. However, the adjective "irrational" is not generally used for functions.

The rational function ${\displaystyle f(x)={\tfrac {x}{x}}}$ is equal to 1 for all x except 0, where there is a removable singularity. The sum, product, or quotient (excepting division by the zero polynomial) of two rational functions is itself a rational function. However, the process of reduction to standard form may inadvertently result in the removal of such singularities unless care is taken. Using the definition of rational functions as equivalence classes gets around this, since x/x is equivalent to 1/1.

The coefficients of a Taylor series of any rational function satisfy a linear recurrence relation, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting like terms after clearing the denominator.

For example,

${\displaystyle {\frac {1}{x^{2}-x+2}}=\sum _{k=0}^{\infty }a_{k}x^{k}.}$

Multiplying through by the denominator and distributing,

${\displaystyle 1=(x^{2}-x+2)\sum _{k=0}^{\infty }a_{k}x^{k}}$

${\displaystyle 1=\sum _{k=0}^{\infty }a_{k}x^{k+2}-\sum _{k=0}^{\infty }a_{k}x^{k+1}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}$

After adjusting the indices of the sums to get the same powers of x, we get

${\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty }a_{k}x^{k}.}$

Combining like terms gives

${\displaystyle 1=2a_{0}+(2a_{1}-a_{0})x+\sum _{k=2}^{\infty }(a_{k-2}-a_{k-1}+2a_{k})x^{k}.}$

Since this holds true for all x in the radius of convergence of the original Taylor series, we can compute as follows. Since the constant term on the left must equal the constant term on the right it follows that

${\displaystyle a_{0}={\frac {1}{2}}.}$

Then, since there are no powers of x on the left, all of the coefficients on the right must be zero, from which it follows that

${\displaystyle a_{1}={\frac {1}{4}}}$

${\displaystyle a_{k}={\frac {1}{2}}(a_{k-1}-a_{k-2})\quad {\text{for}}\ k\geq 2.}$

Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using partial fraction decomposition we can write any proper rational function as a sum of factors of the form 1 / (ax + b) and expand these as geometric series, giving an explicit formula for the Taylor coefficients; this is the method of generating functions.

In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting given a field F and some indeterminate X, a rational expression is any element of the field of fractions of the polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P/Q with Q ≠ 0, although this representation isn't unique. P/Q is equivalent to R/S, for polynomials P, Q, R, and S, when PS = QR. However, since F[X] is a unique factorization domain, there is a unique representation for any rational expression P/Q with P and Q polynomials of lowest degree and Q chosen to be monic. This is similar to how a fraction of integers can always be written uniquely in lowest terms by canceling out common factors.

The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.

Rational functions are used in numerical analysis for interpolation and approximation of functions, for example the Padé approximations introduced by Henri Padé. Approximations in terms of rational functions are well suited for computer algebra systems and other numerical software. Like polynomials, they can be evaluated straightforwardly, and at the same time they express more diverse behavior than polynomials.

Rational functions are used to approximate or model more complex equations in science and engineering including fields and forces in physics, spectroscopy in analytical chemistry, enzyme kinetics in biochemistry, electronic circuitry, aerodynamics, medicine concentrations in vivo, wave functions for atoms and molecules, optics and photography to improve image resolution, and acoustics and sound.

In signal processing, the Laplace transform (for continuous systems) or the z-transform (for discrete-time systems) of the impulse response of commonly-used linear time-invariant systems (filters) with infinite impulse response are rational functions over complex numbers.

• Field of fractions
• Partial fraction decomposition
• Partial fractions in integration
• Function field of an algebraic variety
• Algebraic fractions – a generalization of rational functions that allows taking integer roots

• "Rational function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007), "Section 3.4. Rational Function Interpolation and Extrapolation", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

• Dynamic visualization of rational functions with JSXGraph