rationing function
1. 整有理函数
entire modular form 整模形式entire rational function 整有理函数entire series 整级数
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1. 有理函数域
field of numbers 数域field of rational functions 有理函数域field of rationals 有理数域
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1. 有理分式函数
rational fraction 有理分式rational fractional function 有理分式函数rational function 有理函数
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1. 有理函数
英语词汇-数学词汇_专业词汇 ... rational expression 有理式；有理数式 rational function 有理函数 rational index 有理数指数.
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rationing function
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一些数学函数词汇的意思power functionrational functiontrigonometric function
一些数学函数词汇的意思power functionrational functiontrigonometric function
幂函数,对数函数,三角函数
功率函数有理函数三角函数[From Encyclopedia of Mathematics
A rational function is a function , where
is rational expression in , i.e. an expression obtained from an independent variable
and some finite set of numbers (real or complex) by means of a finite number of arithmetical operations. A rational function can be written (non-uniquely) in the form
are polynomials, . The coefficients of these polynomials are called the coefficients of the rational function. The function
is called irreducible when
have no common zeros (that is,
are relatively prime polynomials). Every rational function can be written as an irreducible fraction ; if
has degree
has degree , then the degree of
is either taken to be the pair
or the number
A rational function of degree
with , that is, a , is also called an entire rational function. Otherwise it is called a fractional-rational function. The degree of the rational function
is not defined. When , the fraction
is called proper, and it is called improper otherwise. An improper fraction can be uniquely written as
is a polynomial, called the integral part of the fraction , and
is a proper fraction. A proper fraction, , in irreducible form, where
admits a unique expansion as a sum of simple partial fractions
is a proper rational function with real coefficients and
are real numbers such that
for , then
can be uniquely written in the form
where all the coefficients are real. These coefficients, like the
in (1), can be found by the method of indefinite coefficients (cf. ).
A rational function of degree
in irreducible form is defined and analytic in the extended complex plane (that is, the plane together with the point ), except at a finite number of singular points, poles: the zeros of its denominator and, when , also the point . Note that if , the sum of the multiplicities of the poles of
is equal to its degree . Conversely, if
is an analytic function whose only singular points in the extended complex plane are finitely many poles, then
is a rational function.
The application of arithmetical operations (with the exception of division by ) to rational functions again gives a rational function, so that the set of all rational functions forms a field. In general, the rational functions with coefficients in a field form a field. If ,
are rational functions, then
is also a rational function. The derivative of order
of a rational function of degree
is a rational function of degree at most . An indefinite integral (or primitive) of a rational function is the sum of a rational function and expressions of the form . If a rational function
is real for all real , then the indefinite integral
can be written as the sum of a rational function
with real coefficients, expressions of the form
and an arbitrary constant
(where , , ,
are the same as in (2), and ,
are real numbers). The function
can be found by the , which avoids the need to expand
into partial fractions (2).
For ease of computation, rational functions can be used to approximate a given function. Attention has also been paid to rational functions
in several real or complex variables, where
are polynomials in these variables with , and to abstract rational functions
are linearly independent functions on some compact space , and
are numbers. S .
I.I. [I.I. Privalov] Priwalow,
"Einführung in die Funktionentheorie" , 1–3 , Teubner
(Translated from Russian)[2]
A.G. Kurosh,
"Higher algebra" , MIR
(Translated from Russian)
For approximation results see .
J.B. Conway,
"Functions of one complex variable" , Springer
(1973)[a2]
"Algebra" , Addison-Wesley
Rational functions on an algebraic variety are a generalization of the classical concept of a rational function (see section 1)). A rational function on an irreducible
is an equivalence class of pairs , where
is a non-empty open subset of
on . Two pairs
are said to be equivalent if
on . The rational functions on
form a field, denoted by .
In the case when
is an irreducible , the field of rational functions on
coincides with the field of fractions of the ring . The transcendence degree of
is called the dimension of the variety .
I.R. Shafarevich,
"Basic algebraic geometry" , Springer
(Translated from Russian)
Vik.S. Kulikov
How to Cite This Entry: Rational function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rational_function&oldid=17805This article was adapted from an original article by E.P. Dolzhenko (originator),
which appeared in Encyclopedia of Mathematics - ISBN .rational function是什么意思_百度知道
rational function是什么意思
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rational function
n.有理函数满意的话请点击“满意”【采纳】
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出门在外也不愁rational function rational function 是怎么定义的_百度知道
rational function rational function 是怎么定义的
提问者采纳
您说的应该是有理函数,就不是多项式,只要判断上下是否分别为多项式就好，有理函数是指任何能被写成两个多项式之比的函数,x出现在分母里,或x出现在根号里.因此.最简单的两条规则您好
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出门在外也不愁}