The earliest definition of parity function
In 1727, [1] Young Swiss mathematician Euler In a paper (originally in Latin) submitted to the St. Petersburg Academy of Sciences to solve the "anti ballistic problem", the concepts of odd and even functions were proposed for the first time. If - x is used instead of x, and the function remains unchanged, such function is called even function (Latin functions pares). Euler listed three categories Even function And three categories Odd function The properties of odd even functions are discussed. French mathematician Darumbel (J.R.D. Alembert, 1717-1783) Diderot (D. Diderot, 1713-1784) The entry on function in Volume 7 of Encyclopedia (published in 1757), edited by the chief editor, said: "Ancient geometers, more precisely, ancient analysts, called different powers of a certain quantity x functions of x." Similarly, French mathematicians Lagrange At the beginning of Analytic Function Theory (1797), it was also said that early analysts used“ function ”This word only means "different powers of the same quantity". Later, its meaning was extended to mean "all quantities derived from other quantities in any way", Leibniz And John Bernoulli first adopted the latter meaning. In his paper in 1727, Euler did not involve anything in discussing odd and even functions Transcendental function 。 Therefore, the earliest concepts of odd and even functions are aimed at power function And related Composite function As far as Euler is concerned, the names of "odd function" and "even function" are obviously derived from the parity of the exponent or exponential molecule of power function: power function with even exponent is even function, and power function with odd exponent is odd function. The Concept of Odd and Even Functions in Introduction to Infinite Analysis
In 1748, [1] Euler published his famous mathematical book Introduction to Infinite Analysis, which established function as the most basic research object of analysis. In the first chapter, he gave the definition of function, classified functions, and again discussed two special types of functions: even function and odd function. Euler's definitions of odd and even functions are essentially the same as those in his 1727 paper, but he discussed more types of odd and even functions and gave more properties of odd functions.
Euler's Puzzlement and Mistakes
Euler believes that the function And function It is equivalent, so although the product of odd function and even function is odd function, sometimes such product may also be even function. In view of this, Euler proposed that if the power of an even function is still an even function, the power exponent must be limited. In particular, if the exponent is a fraction, then its denominator cannot be an even number. In defining even functions as and Euler added a constraint condition in particular: Cannot contain Radicals like that [1] 。 Obviously, Euler failed to distinguish between functions Sum function 。 Parity functions in French and English
Although D'Alembert gave the definition of function in Encyclopedia and introduced Rational function 、 Irrational function 、 Homogeneous function , similar functions, but not "odd function" or "even function". In 1786, F. Pezzi, a Frenchman, translated Volume 1 of Introduction to Infinite Analysis into French, and "odd function" and "even function" were translated into "function pay" and "function pay" respectively, which was the first time that two mathematical nouns appeared in French.
French mathematician in 1792 legendre (A. Legendre) (1752-1833) proposed "even function of sine function" in his paper "On the transcendence of ellipse" submitted to the Academy of Sciences. Legendre may have followed Peiqi's translation or directly translated Euler's name. What we need to point out here is that translating the Latin of "even function" and "odd function" into corresponding French will not produce different translations, because no later than Descartes (R. Descartes, 1596-1650) already has the names of "even" (nombres pairs) and "odd" (nombres pairs) in French in Geometry. The names "odd function" and "even function" were not widely used in France at the end of the 18th century; In other words, the parity of functions has not yet received widespread attention from French mathematicians at that time. In 1796, French mathematician Rabe translated the whole book Introduction to Infinite Analysis into French, in which Rabe also translated "odd function" and "even function" into "function pay" and "function pay" respectively
In 1809, the Scottish mathematician W. Wallace (1768-1843) translated Legendre's paper into English and published it in the Mathematics Repository. Wallace naturally translated "function paire" into "even function". This is the first time that the word "even function" appears in the English world. However, famous mathematicians in Britain Hutton (C. Hutton, 1737-1823) In the Dictionary of Mathematics and Philosophy published in 1815, although there are two entries of "function" and "function in calculus", the odd and even functions are ignored. and DeMorgan The Foundation of Algebra (translated by Wei Lie Yali and Li Shanlan into Algebra) clearly classifies functions, but still does not mention odd or even functions. In the United States, the best-selling book of calculus, Analytic Geometry and Fundamentals of Calculus (translated by Li Shanlan and Wiley Yali as "Substituting for Differential Accumulation and Gradation") by mathematician E. Loomis (1811-1889) gives the names of implicit function, explicit function, increasing function and decreasing function, but it also does not contain odd and even functions. This shows that the concepts of odd and even functions and the new nouns introduced by Wallis have not yet been widely spread and widely concerned in the English world in the first half of the 19th century. Accordingly, the two concepts are not found in the western mathematical translations of the late Qing Dynasty in China. It was not until the early 20th century that the two concepts were introduced into China. The Glossary of Mathematical Nouns published in 1938 and Mathematical Nouns published in 1945 both contain two nouns. 1. If you know the function expression, for the Define Fields Any x in it satisfies f (x)=f (- x), such as y=x * x; 2. If you know the image, the even function image is about Y-axis (Line x=0) Symmetric
3. Definition Domain D About origin Symmetry is a necessary and insufficient condition for this function to become even For example, if f (x)=x ^ 2, x ∈ R, then f (x) is an even function. f (x)=x ^ 2, x ∈ (- 2,2]) (f (x) is equal to the square of x, - 2<x ≤ 2), then f (x) is not an even function
Number.
It is mainly based on the definition of odd and even functions to determine whether the definition domain is Origin of Symmetry , if it is asymmetric, it is Non odd and non even , if symmetric, f (- x)=- f (x) is an odd function; F (- x)=f (x) is an even function [2] 。
About Origin The symmetric function is Odd function , about Y axis Symmetric functions are even functions. If f (x) is an even function, then f (x+a)=f [- (x+a)]
But if f (x+a) is an even function, then f (x+a)=f (- x+a)
(1) The sum of two even functions and Even function [3] .
(2) Two odd The sum of functions is odd Function
(3) One even Function and a odd The sum of functions is Nonsingular Function and Non even Function
(4) Multiplied by two even functions product It is an even function
(5) Two Odd function The product obtained by multiplication is Even function . (6) One Even function With a Odd function Multiplied product by Odd function .
(7) Odd function must satisfy f (0)=0 (Because the expression F (0) means that 0 is within the scope of the definition domain, F (0) must be 0) Not necessarily odd function has f (0), But there are F (0) must be equal to 0 when F (0) , f (0)=0 is not necessary, and odd function is derived. At this time, the function is not necessarily odd, for example, f (x)=x ^ 2
(8) Defined in R The odd function f (x) on must satisfy f (0)=0; because Define Fields On R, there is f (0) at the point x=0 origin Symmetry, only one y value can be taken at the origin, and only f (0)=0. This is an article that can be used directly conclusion : When x can be taken as 0 and f (x) is an odd function, f (0)=0). (9) If and only if f (x)=0 (the domain is symmetric about the origin), f (x) is both odd and even functions.
(10) On the symmetric interval, the integrand is odd definite integral Is zero. 
