"A function is a quantity obtained from another quantity through a series of algebraic operations."
The above definition is obviously too narrow, because it actually only applies to the scope of algebraic functions. Therefore, in the subsequent development, the concept of function has been further expanded. With the deepening of mathematical research, people gradually come into contact with some transcendental functions, such as logarithmic function and trigonometric function of exponential function. Although these functions are beyond the scope of algebraic functions, in the eyes of some mathematicians, the difference between them is that they repeat the operations of algebraic functions infinitely many times, so people put forward the following through weak abstraction.
"A function is an analytic expression formed by a variable and some constants in any way (finite or infinite)."
This definition given by Euler, though still too narrow, was still dominant for a long time in18th century.
/kloc-At the beginning of the 9th century, the concept of function was expanded again, and the concept of function began to get rid of "analytic expression". In addition, Dirichlet put forward the following concept of function:
If there is a unique Y value corresponding to each X value in a given interval, then Y is a function of X..
Finally, if any mathematical object is used instead of specific quantity and the language of * * * theory is adopted, a more general concept of "mapping" can be obtained:
If there is a definite correspondence between two * * * elements, it is called mapping.
The mathematical term function is used by Leibniz in 1694 to describe a related quantity of a curve, such as the slope of the curve or a certain point on the curve.
The function that Leibniz refers to is now called derivative function, and the functions that ordinary people except mathematicians generally come into contact with belong to this category.
For a differentiable function, we can discuss its limit and derivative.
Both of them describe the relationship between the change of function output value and the change of function input value, and are the basis of calculus.
In 17 18, johann bernoulli defined a function as "a function of a variable refers to a quantity composed of this variable and a constant in any way." 1748, leonhard euler, a student in johann bernoulli, said in his book Introduction to Infinite Analysis: "The function of variables is an analytical expression composed of variables and some numbers or [constants] in any way".
For example, f(x) = sin(x)+x3.
1775, Euler put forward the definition of function in the book Principles of Differential Calculus: "If some quantities depend on other quantities in the following way, that is, when the latter changes, the former itself changes, then the former quantity is called the function of the latter quantity."
/kloc-mathematicians in the 0 th and 9 th centuries began to standardize all branches of mathematics.
Karl Weierstrass proposed that calculus should be based on arithmetic, not geometry, so he preferred Euler's definition.
By extending the definition of function, mathematicians can study some "strange" mathematical objects, such as non-derivative continuous functions.
These functions were once considered to have only theoretical value, but they were still regarded as "monsters" until the beginning of the 20th century.
Later, people found that these functions played an important role in the modeling of physical phenomena such as Brownian motion.
By the end of 19, mathematicians began to try to standardize mathematics with * * * theory.
They tried to define each mathematical object as a * * *.
Johann peter gustav lejeune dirichlet gave the modern formal definition of function.
Dirichlet's definition regards function as a special case of mathematical relationship.
But for practical application, the difference between modern definition and Euler definition can be ignored.
History shows that important mathematical concepts have an inestimable effect on the development of mathematics, and the influence of function concepts on the development of mathematics can be said to be lasting and extraordinary throughout the ages. Looking back at the historical development of the concept of function, it is a very beneficial thing to look at the historical process in which the concept of function has been continuously refined, deepened and enriched, which not only helps us to improve the clarity of understanding the context of the concept of function, but also helps us to understand the influence of mathematical concepts on mathematical development and learning.
( 1)
Marx once thought that the concept of function originated from the study of indefinite equations in algebra. Since Diophantine in Roman times had studied indefinite equations, the concept of function had sprouted at least then.
Since Copernicus's astronomical revolution, sports have become a common problem for Renaissance scientists. People are thinking: since the earth is not the center of the universe, it has its own rotation and revolution, why does the falling body fall vertically to the earth instead of deflecting? The orbit of this planet is elliptical. What is the principle? In addition, the research on the flight route, range and reachable height of projectiles on the earth's surface and the influence of projectile speed on height and range are not only problems that scientists try to solve, but also problems that military strategists demand to solve. The concept of function is a mathematical concept derived from the study of motion, and motion is the mechanical source of the concept of function.
(2)
Long before the concept of function was clearly put forward, mathematicians had contacted and studied many specific functions, such as logarithmic function, trigonometric function, hyperbolic function and so on. Descartes noticed the dependence of one variable on another in Analytic Geometry around 1673, but he didn't realize the need to refine the general concept of function at that time, so Newton and Leibniz didn't establish calculus until 17 century.
1673, Leibniz first used the word "function" to represent "power", and later he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate and tangent length. It can be seen that the original mathematical meaning of the word "function" is quite extensive and vague. Almost at the same time, Newton used another term "flow" to express the relationship between variables in the discussion of calculus until. Swiss mathematician johann bernoulli clearly defined the concept of function on the basis of Leibniz's concept of function. Bernoulli called the quantity formed by variable x and constant in any way "the function of x" and expressed it as yx.
At that time, because the operations connecting variables and constants were mainly arithmetic operations, trigonometric operations, exponential operations and logarithmic operations, Euler simply named the formula connecting variables X and constants C as analytic functions and divided them into "algebraic functions" and "transcendental functions".
/kloc-in the middle of the 8th century, D'Alembert and Euler introduced the concept of "arbitrary function" successively because of the study of string vibration. When D'Alembert explained the concept of "arbitrary function", he said that it meant "arbitrary analytical formula", while Euler thought it was "arbitrarily drawn curve". Now, it seems that these are all expressions of functions and an extension of the concept of functions.
(3)
The lack of scientific definition of the concept of function has caused a sharp contradiction between theory and practice. For example, partial differential equations are widely used in engineering technology, but the lack of scientific definition of functions greatly limits the establishment of partial differential equation theory. From 1833 to 1834, Gauss began to turn his attention to physics. In the process of inventing the telegraph with W Wilbur, he did a lot of magnetic experiments. The important theory that "the force is inversely proportional to the square of the distance" makes function appear as an independent branch of mathematics, which actually needs to prompt people to further study the definition of function.
Later, people gave a definition: if one quantity depends on another quantity, when the latter quantity changes, the former quantity also changes, then the first quantity is called the function of the second quantity. "Although this definition has not revealed the essence of the function, it is a gratifying progress to inject changes and movements into the definition of the function."
In the history of function concept development, the work of French mathematician Fourier has the greatest influence. Fourier profoundly reveals the essence of functions, and thinks that functions need not be limited to analytic expressions. 1822, he said in his masterpiece Analytical Theory of Heat, "Usually, a function represents a set of connected values or vertical coordinates, each of which is arbitrary ... and we don't assume that these vertical coordinates obey a * * *. In every way, they are adjacent. " In this book, he expressed a function given by discontinuous "lines" in the form of sum of trigonometric series. More precisely, any periodic function with 2π can be expressed by [-π, π].
? It shows that among them,
Fourier's research fundamentally shook the old traditional thought about the concept of function, which caused great shock in the mathematics field at that time. There is no insurmountable gap between analytic formula and curve. Series links analytic formula with curve, and the view that function is analytic formula eventually becomes a huge obstacle to reveal the relationship between functions.
Through an argument, the function definitions of Lobachevsky and Dirichlet came into being.
1834, Russian mathematician Lobachevsky put forward the definition of function: "The function of x is such a number that it has a definite value for each x and changes with it. The function value can be given by an analytical formula or a condition, which provides a method to find all the corresponding values. This dependence of the function can exist, but it is still unknown. " This definition establishes the correspondence between variables and functions.
1837, the German mathematician Dirichlet thought that how to establish the relationship between X and Y was not important, so his definition was: "If for every value of X, Y always has a completely certain value corresponding to it, then Y is a function of X."
According to this definition, even if it is expressed as follows, it is still said to be a function (Dirichlet function):
f(x)= 1？ (x is a rational number),
0? X is an irrational number.
In this function, if the value of x increases gradually from 0, then f(x) will suddenly change from 0 to 1. In any cell, f(x) will suddenly change from 0 to 1 infinitely. Therefore, it is difficult to express it with one or several formulas, and even whether an expression can be found is a problem. But whether it can be expressed by expression is stated in Dirichlet.
Dirichlet's definition of function avoids all the descriptions of dependence in the previous definition of function, and is unconditionally accepted by all mathematicians in a completely clear way. At this point, it can be said that the concept of function and the essential definition of function have been formed, which is the classical definition of function that people often say.
(4)
The further development of production practice and scientific experiments has caused new sharp contradictions in the concept of function. In the 1920s, people began to study microphysical phenomena. 1930 quantum mechanics came out, and a new function-δ function was needed in quantum mechanics.
Namely. ρ(x)= 0，x≠0，
∞，x=0。
and
The appearance of δ-function has caused a heated debate. According to the original definition of function, only the correspondence between numbers is allowed, and "∞" is not regarded as a number. In addition, it is incredible that there is only one function whose point is not zero for the independent variable, but its integer value is not equal to zero. However, the δ-function is really an abstraction of the actual model. For example, cars and trains cross the bridge, which is natural.
P(0)= pressure/contact surface = 1/0 = ∞.
At the rest point x≠0, there is no pressure because there is no pressure, that is? P(x)=0。 In addition, we know that the integral of the pressure function is equal to the pressure, that is
? Under such historical conditions, the concept of function develops actively, resulting in a new modern function definition: if any element X of *** M always has an element Y determined by *** N corresponding to it, it is said that a function is defined on *** M, and it is recorded as y=f(x). Element x is called independent variable and element y is called dependent variable.
Although there is only one word difference between the modern definition of function and the classical definition in form, it is a major development in concept and a major turning point in the development of mathematics. Modern functional analysis can be used as a sign of this turning point, and it studies functional relations in a general sense.
After more than 200 years of tempering and transformation, the definition of function has formed a modern definition of function, which should be said to be quite perfect. However, the development of mathematics is endless, and the formation of modern definition of function does not mean the historical end of the development of function concept. In the past twenty years, mathematicians have attributed functions to a broader concept-"relationship".
Let *** X and y, and we define the product set X×Y of x and y as
X×Y={(x，y)|x∈X，y∈Y}。
A subset of the product set X×Y, r is called the relationship between x and y, and if (x, y)∈R, it is said that x has a relationship with y, which is denoted as xRy. If (x, y)R, say that x and y have nothing to do.
Let's assume that F is the relationship between X and Y, that is, fX×Y Y Y. If (x, y), (x, z)∈f, there must be y=z, then F is called a function from X to Y. In this definition, the term "correspondence" has been avoided formally, and all languages of * * * theory have been used.
From the whole process of the development of the above-mentioned function concepts, we realize how important it is to study, explore and broaden the connotation of mathematical concepts by connecting with reality and consulting a large number of mathematical materials.