Function, a mathematical term.Its definition is usually divided into traditional definition and modern definition. The essence of the two definitions of function is the same, but the starting point of the narrative concept is different. The traditional definition is from the point of view of motion change, while the modern definition is from the point of view of set and mapping.The modern definition of function is to give a number set A, assume that the element in it is x, apply the corresponding rule f to the element x in A, recorded as f (x), and get another number set B. If the element in B is y, then the equal relationship between y and x can be expressed by y=f (x). The concept of function contains three elements: definition field A, value field B, and corresponding rule f.The core is correspondence rule f, which is the essential feature of functional relations.[1]
Function, first developed by Chinese mathematicians in the Qing DynastyLi ShanlanTranslation is based on his book Algebra.The reason why he translated this way is that "where this variable functions another variable, this is the function of that variable", that is, function means that one quantity changes with the change of another quantity, or that one quantity contains another quantity.
First, understand that a function is a corresponding relationship between sets.Then, understand theFunctional relationThere is more than one.Finally, we should focus on understanding the three elements of the function.
The correspondence rule of function is commonly usedAnalytic expressionHowever, a large number of functional relationships cannot be expressed in analytic expressions, and can be expressed in images, tables and other forms[2]。
concept
In a change process, the variable that changes is called a variable (in mathematics, the variable is x, and y changes with the value of x). Some values do not change with the variable, and they are called constants.
independent variable(Function): A variable associated with another quantity. Any value of this quantity can find a corresponding fixed value in the other quantity.
dependent variable(Function): It changes with the change of independent variable, and when the independent variable takes a unique value, the dependent variable (function) has and only has a unique value corresponding to it.
Function value: in a function where y is x, x determines a value, and y then determines a value. When x takes a, y then determines b, and b is called a function value[2]。
Mapping Definitions
Let A and B be two non empty sets
For any element a in set A, there is a unique element b corresponding to it in set BaggregateThe corresponding relationship between A and set B f) is called set A to set Bmapping(Mapping), recorded as
。Where, b is called the image of a under the mapping f, which is recorded as:
；A is called b with respect to the mapping fPrimal image。The set of images of all elements in set A is marked f (A).
The mapping defined between non empty number sets is called function.(The independent variable of a function is a special primitive,dependent variableIs a special image)[2]
Geometric meaning
Function andInequalityandequationThere is a connection（Elementary function）。Let the value of the function equal to zero. From a geometric perspective, the value of the corresponding independent variable is the abscissa of the intersection point of the image and the X axis;From the perspective of algebra, the corresponding independent variable isSolution of the equation。In addition, the function'sexpression(except for functions without expressions), replace "=" with "<" or ">", and then replace "Y" with othersAlgebraic expression, the function becomes an inequality, and the range of independent variables can be found[2]。
set theory
If the binary relationship from X to Y
, for each
, there is only one
, making
, then f is called the function from X to Y, recorded as:
。
When
F is called n-variable function[2]。
element
Set of input valuesXgo by the name offDomain of definition;Set of possible output valuesYgo by the name offOfrange。The value range of a function means that all elements in the definition domain are mappedfThe set of actual output values obtained.Note that it is incorrect to call the corresponding field the value field. The value field of a function is a subset of the corresponding field of a function.
In computer science, the data types of parameters and return values determinesubroutineDefinition field and corresponding field of.Therefore, the definition domain and corresponding domain are mandatory constraints determined at the beginning of the function.On the other hand, the value range is related to the actual implementation[2]。
classification
Single shot, full shot, double shot
MonomorphismFunction to map different variables to different values.That is, for all
and
, when
happen now and then
。
SurjectionFunction, whose value field is its corresponding field.That is, for any y in the corresponding field of mapping f, there is at least one x that satisfies y=f (x).
BirefringenceFunction is both injective and injective.It is also called one-to-one correspondence.Bijective functions are often used to indicate that sets X and Y areEquipotentialThat is, they have the same cardinality.If a one-to-one correspondence can be established between two sets, the two sets are said to be equipotential[2]。
Image and primal image
The element in the image is f (x), and their value is 0[2]。
image
functionfThe image of is a point pair on the plane
Where x is taken from all members of the domain.Function graphs can help to understand and prove some theorems.
IfXandYIf they are all continuous lines, the graph of the function has a very intuitive expression. Pay attention to the two setsXandYThere are two definitions for the binary relationship of（X,Y,G）, whereGIs a graph of relationships;The other is simply defined by relational graph.Use the second definition to define the functionfEqual to its image[2]。
Development history
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Origin of function
The word "function" used in Chinese math books is a translation word.It was Li Shanlan, a mathematician in the Qing Dynasty, who translated "function" into "function" when translating Algebra (1859).
In ancient China, the word "letter" and the word "contain" are common, both of which have“contain”Means.Li Shanlan's definition is: "Where contains heaven, it is a function of heaven." In ancient Chinaday"," land "," people "and" things "unknown numberOr variables.The meaning of this definition is: "Where a formula contains a variable x, the formula is called a function of x." So "function" means that a formula contains a variable.The exact definition of an equation isequation。howeverequationA monograph on mathematics in the early period of China《Chapter Nine Arithmetic》In, it means simultaneous linear equations containing multiple unknowns, that isLinear equations[2]。
Early concepts
In the seventeenth century, Galileo Galilei's book Two New Sciences almost all contained the concept of function or variable relationship, and expressed the relationship between functions in words and proportions.Around 1637Descartes In his analytic geometry, he had noticed the dependence of one variable on another, but he did not realize that the concept of function should be refined at that time, so untilseventeenth centurylater stageNewton、LeibnizestablishCalculusAt that time, no one knew the general meaning of functions. Most functions were studied as curves.
In 1673, Leibniz first used "function" to express“power”Later, he used the word to expressspotOfAbscissa、Ordinate、Tangent lengthRelation of points on isocurveGeometric quantity。meanwhile,NewtonIn the discussion of calculus“flow”To represent the relationship between variables[2]。
eighteenth century
In 1718,Johann Bernoulli stayLeibnizOn the basis of the concept of function, the concept of function is defined: "quantity composed of any form of any variable and constant." He means that all formulas composed of variable x and constant are called functions of x, and emphasizes that functions should be expressed in formulas.
In 1748,EulerIn his book Introduction to Infinite Analysis, he defined function as: "The function of a variable is an analytical expression composed of some numbers or constants of the variable and any way." He called the function definition given by John Bernoullianalytic functionAnd further distinguish it into algebraic functions andTranscendental function, "random function" is also considered.It is not difficult to see that the definition of function given by Euler is more general and more extensive than that of John Bernoulli.
In 1755, Euler gave another definition: "If some variables depend on other variables in a certain way, that is, when the latter variables change, the former variables also change, and the former variables are called functions of the latter variables."[2]
nineteenth century
In 1821,CauchyThe definition is given from the definition of variables: "There is a certain relationship between some variables. Once the value of one variable is given, the value of other variables can be determined along with it, then the original variable is called independent variable, and the other variables are called functions." In Cauchy's definitionindependent variableAt the same time, it points out that there is no need for analytic expressions for functions.But he still thinksFunctional relationMultipleAnalytic expressionThis is a big limitation.
1822Fourier It is found that some functions can be expressed in curves, in one formula, or in multiple formulas, thus ending the debate on whether the concept of function is expressed in a single formula, and advancing the understanding of function to a new level.
1837DirichletBreaking through this limitation, how to establish
And
The relationship between them is irrelevant. He expanded the concept of function and pointed out that:“For every definite x value in an interval, y has a definite value, so y is called a function of x.”This definition avoids the description of dependencies in function definitions and is accepted by all mathematicians in a clear way.This is what people often sayDefinition of classical function。
wait untilCantorAfter the established set theory occupied an important position in mathematics,Oswald WibranThe modern function definition is given with the concepts of "set" and "correspondence". Through the concept of set, the corresponding relationship, definition domain and value domain of function are further specified, and the limit of "variable is number" is broken. Variables can be numbers or other objects[2]。
Modern concepts
1914Hausdorff(F. Hausdorff) defined the function with the ambiguous concept "order pair" in the Outline of Set Theory, which avoids the ambiguous concepts of "variable" and "correspondence".Kuratovsky(Kuratowski) used in 1921Set conceptTo define "ordinal couple" makes Hausdorff's definition very precise.
In 1930, the new modern function was defined as“If for any element x of set M, there is always an element y determined by set N corresponding to it, then it is called to define a function on set M, which is recorded as f.elementX is calledindependent variable, the element y is calleddependent variable”[2]。
Function definition
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Traditional definition
In general, in a change process, if there are two variables x and y, if there is a unique and definite y corresponding to any x, then x isindependent variableY is a function of x.The value range of x is called the definition field of the function, and the corresponding value range of y is called the function'srange[2]。
Modern definition
Let A and B be non empty number setsaggregateAny number x in A has a unique number in set B
be calledCorrespondence rule。The definition field, value field and corresponding rule are calledThree elements of function
Definition field, value field and corresponding rule are called the three elements of function.Generally written as
。If the definition field is omitted, it generally means that theaggregate[2]。
programming
These statements in function procedures are used to accomplish some meaningful work - usually processing text, controlling input or calculating values.By introducing the function name and required parameters into the program code, you can execute (orcall）This function.
Similar process, but the function generally has aReturn value。They can call themselves in their own structure, which is called recursion.
Most programming languages have functions in their methods of building functionskeyword(or calledReserved word）[2]。
Representation method
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Analytic method
The method of expressing the functional relationship between two variables with equations containing mathematical relations is called analytic expression method.The advantage of this method is that it can express the function andindependent variableThe quantitative relationship between them;The disadvantage is that the corresponding value is often calculated through more complex operations, and some functional relationships in practical problems may not be expressed in expressions[2]。
Tabulation method
The list method is used to express the functional relationship between two variables.The advantage of this method is that the corresponding function value can be read directly through the known independent variable value in the table;The disadvantage is that only part of the corresponding values can be listed, which is difficult to reflect the full picture of the function.As shown below[2]：
x
one
two
three
four
y=2x
two
four
six
eight
Image method
Take the value of the independent variable x and the corresponding dependent variable y of a function as the abscissa and ordinate of the point respectively, and draw its corresponding point in the rectangular coordinate system. The graph formed by all these points is called the image of the function.This method of representing functional relations is called graph method.The advantage of this method is that the function relation can be expressed intuitively and vividly through the function image;The disadvantage is that the quantitative relationship obtained from image observation is approximate[2]。
Language narration
Use language to describe the relationship between functions[2]。
Properties of functions
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Boundedness
Let the function f (x) be defined on the interval X. If M>0 exists, all the functions belonging tosectionX on X is always | f (x) | ≤ M, then it is said that f (x) is bounded on the interval X, otherwise it is said that f (x) is unbounded on the interval[3]。
Monotonicity
Let the definition field of function f (x) be D,sectionI is contained in D.If for any two points x on the intervaloneAnd xtwo, when xone<xtwoWhen, there is always f (xone）<f（xtwo）, then the function f (x) is monotone on the interval IIncrementalOf;For any two points x on interval IoneAnd xtwo, when xone<xtwoWhen, there is always f (xone）>f（xtwo）, then the function f (x) is monotonically decreasing on the interval I.The monotonically increasing and monotonically decreasing functions are collectively calledMonotone function[2]。
Parity
set up
Is a real valued function of a real variable. If f (- x)=- f (x), thenf（x）ForOdd function。
In geometry, an odd function is symmetric about the origin, that is, its image will not change after rotating 180 degrees around the origin.
Examples of odd functions are x, sin (x), sinh (x), and erf (x).
set upf（x）It is a real valued function of a real variable, if any
The domain D of is an unbounded interval on at least one side. If D is bounded, the function is not periodic.Not every periodic function has a minimum positive period, such asDirichlet function。
Periodic functions have the following properties:
(1) If T (T ≠ 0) is the period of f (x), then - T is also the period of f (x);
(2) If T (T ≠ 0) is the period of f (x), then nT (n is any non-zero integer) is also the period of f (x);
(3) If T1 and T2 are both cycles of f (x), then
Is also the period of f (x);
(4) If f (x) has the minimum positive period T *, then any positive period T of f (x) must be a positive integer multiple of T *;
(5) T * is the minimum positive period of f (x), and T1 and T2 are two periods of f (x) respectively, then T1/T2 ∈ Q (Q is a rational number set);
(6) If T1 and T2 are two cycles of f (x), and T1/T2 is an irrational number, then f (x) does not have a minimum positive cycle;
(7) The domain M of periodic function f (x) must be an unbounded set of both sides.[2]
Continuity
In mathematics, continuity is an attribute of a function.Intuitively speaking, a continuous function is a function whose output change will be small enough when the change of input value is small enough.If a small change in the input value will produce a sudden jump in the output value or even can not be defined, this function is called a discontinuous function (or discontinuous).
set upfIs aSet of real numbersThe function that a subset of f hits: a point incIs continuous if and only if the following two conditions are met:
fAt pointcThere are definitions on.C is one ofOne ofAccumulation point, and regardless of the argumentxHow to approach inc，f（x）OflimitBoth exist and equal tof（c）。A function is said to be continuous everywhere or everywhere, or simply continuous, if it is continuous at any point in its domain.More generally, a function in thesubsetUpper is continuous when it is continuous at every point of the subset.
Without the concept of limit, the continuity of real valued function can also be defined by the following so-called method.
Still consider the function.hypothesiscyesfElement in the domain of the.functionfIs known to becPoint continuityif and only ifThe following conditions are true:
For any positive real number, there is aPositive real numberδ> 0 so that for δ in any domain, as long asxsatisfyc - δ< x < c + δ，Is established[2]。
There is a definition on D (D is the definition domain of composite function, which can be
A non empty subset of the domain), and
, then function
Is called a function
Sum function
The definition domain of the composite function is D, and the variable is
It is called an intermediate variable.
Not any two functions can be combined into a composite function. If D is an empty set, then
Sum function
Cannot compound[3]。
Inverse function
In general, let the function
, the value field is W, for each y belonging to W, there is a unique x belonging to D, so that f (x)=y, then the variable x is also a function of the variable y, called the inverse function of y=f (x), recorded as
。The inverse function of y=f (x) is conventionally recorded as
。
Traditionally, only one-to-one corresponding functions have inverse functions.If a function is monotonically increasing or decreasing defined in its domain D, its inverse function monotonically increases or decreases in its domain W.Symmetry between primitive function and inverse function with respect to y=x[3]。
Piecewise function
A function whose corresponding rules are expressed in different analytic formulas within different variation ranges of independent variables is called piecewise function[3]。The definition domain of piecewise function isUnion[2]。
Polynomial function
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Constant function
When x is any number in the definition field, there is y=C (C is a constant), then the function y=C is called a constant function,
Its image is a straight line or part of a straight line parallel to the x-axis[2]。
Linear function
In a certain change process, set two variables x and y, if you can write
In y=kx+b (k, b is a constant, k ≠ 0), when x increases m, the function value y increases km; on the contrary, when x decreases m, the function value y decreases km.
2. When x=0, b is the intersection of the linear function image and the y-axisOrdinate, the coordinates of this point are (0, b); when y=0, the image of the primary function intersects the x-axis at (- b/k)
3. When b=0, the primary function becomes a positive proportional function.Of course, the positive proportional function is a special linear function.
4. In two linear function expressions:
When k and b in the expressions of two linear functions are the same, the images of these two linear functions coincide;
When k in the expressions of two linear functions is the same and b is different, the images of these two linear functions are parallel;
When k and b in the expressions of two linear functions are different, the images of these two linear functions intersect;
When k in the expressions of two linear functions is different and b is the same, then the images of these two linear functions intersect at the same point on the y-axis (0, b);
When k in two linear function expressions is negative reciprocal to each other, the two linear function images are perpendicular to each other.
5. When two linear functions (y1=k1x+b1, y2=k2x+b2) are multiplied (k ≠ 0), the new function obtained is a quadratic function,
When k1 and k2 are positive and negative, the opening of the quadratic function is upward;
When the positive and negative of k1 and k2 are opposite, the opening of the quadratic function is downward.
The intersection point of quadratic function and y-axis is (0, b2b1).
6. The ratio of two linear functions (y1=ax+b, y2=cx+d), the new function y3=(ax+b)/(cx+d) obtained is an inverse proportional function, and the asymptote is x=- b/a, y=c/a.
7. WhenRectangular coordinate system When two lines in the middle are parallelFunction analytic expressionThe values of k in (i.e. the coefficient of the first order term) are equal;When two straight lines in the plane rectangular coordinate system are perpendicular, the values of k in the function analytic formula are mutuallyNegative reciprocal(That is, the product of two k values is - 1).
Image:
image of linear function
As shown in the right figure, the linear function y=kx+b (k ≠ 0) image is a straight line, passing through two points (0, b) and (- b/k, 0).In particular, when b=0, the image crosses the origin.
Relation and difference between linear function and equation:
1. A linear function has a similar expression to a linear equation with one variable.
2. A linear function represents the relationship between a pair of (x, y), and it has countless pairs of solutions;The unary linear equation represents the value of the unknown number x, with only one value at most.
3. The abscissa of the intersection of a linear function and the x-axis is correspondingUnary linear equationThe root of.
onceFunctions and inequalities:
From a functional point of view, the solutionInequalityThe method of is to seek the value range of the independent variable x that makes the value of the primary function y=kx+b greater (or less) than 0;
From the perspective of function image, it is the set formed by determining the abscissa of all points on the upper (or lower) part of the x axis of the line y=kx+b.
The corresponding linear function y=kx+b, its intersection with the x-axis is (- b/k, 0).
When k>0, the solution of inequality kx+b>0 is: x>- b/k, and the solution of inequality kx+b<0 is: x<- b/k;
When the solution of k<0 is: the solution of inequality kx+b>0 is: x<- b/k, and the solution of inequality kx+b<0 is: x>- b/k[2]。
Quadratic function
Quadratic function
Generally, the relationship between the independent variable x and the dependent variable y is as follows:
, then y is called x'sQuadratic function。The definition field of quadratic function is real number field R.The constant term c determines the intersection of the parabola and the y-axis.The parabola intersects the y-axis at (0, c)
The quadratic function can also be expressed in the following two ways:
Vertex:
;
Intersection (with x-axis):
It can be seen from the right figure that the quadratic function image isaxial symmetrygraphical.
Function property
1. The quadratic function isparabolaBut the parabola is not necessarily a quadratic function.A parabola with an opening up or down is a quadratic function.Parabola is an axisymmetric figure.The axis of symmetry is a straight line x=- b/2a.The only intersection between the axis of symmetry and the parabola is the vertex P of the parabola.In particular, when b=0The axis of symmetry is the y-axis(i.e. straight line x=0)
2. The parabola has a vertex P, whose coordinates are
, when
When P is on the y-axis;When
P is on the x-axis.
3. The quadratic coefficient a determines the opening direction and size of the parabola.When a>0, the parabola opens upwards;When a<0, the parabola opens downward|A | The larger, the smaller the opening of the parabola.When a>0, the function
Get minimum value at
；stay
Is a minus function on the
Is an increasing function;The value range of the function is
The opposite is unchanged.
4. The linear term coefficient b and the quadratic term coefficient a jointly determine the position of the axis of symmetry.When a and b are the same sign (that is, ab>0), the symmetry axis is on the left of the y-axis;When a and b are signed differently (that is, ab<0), the axis of symmetry is on the right of the y-axis.
5. Order
, has the following properties:
Δ>0，The parabola has two intersections with the x-axis, namely:
and
。
Δ= 0，The parabola has one intersection with the x-axis, which is
。
Δ<0，The parabola has no intersection with the x-axis, and the value of x isimaginary number[2]。
Cubic function
In the form of
(a ≠ 0, b, c, d are constants) are called cubic functions.CubicimageIt is a curve regression parabola (different from ordinary parabola)[2]。
Quartic function
Definition: like
The function of is called quartic function[2]。
Quintic function
In general, the independent variable x and the dependent variable y have the following relationship:
Y is called the quintic function of x.Wherein, a, b, c, d and e are five, four, three, two and one times respectivelycoefficient, f isconstant，a≠0[2]。
When a>1, it is a strictly monotonic increasing function; when 0<a<1, the function monotonically decreases, and the image passes through a fixed point (0,1)[2]。
Logarithmic function
, called a as the bottom, and the definition field is
, the value field is
。a> 1 is strictly monotonic increase, 0<a<1 is strictly single decrease.Regardless of the value of a, the graph of the logarithmic function passes through the point (1,0), and the logarithmic function and the exponential function are inverse functions of each other.
The logarithm with the base of 10 is called the common logarithm, which is abbreviated as
。In science and technology, the logarithm with e as the base, that is, the natural logarithm, is commonly used as
。
trigonometric function
Trigonometric function belongs to elementary function in mathematicsTranscendental functionA class of functions of.Their essence isArbitrary angleThe mapping between the variables of a set of and a set of ratios.Generaltrigonometric functionIt is defined in the plane rectangular coordinate system, and its definition domain is the whole real number domain.Another definition isright triangleMedium, but not complete.Modern mathematics describes them asinfinite sequence And the solution of differential equation, and extend its definition tocomplexDepartment.
Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single valued function.
Trigonometric functions have important applications in complex numbers.In physics, trigonometric function is also a commonly used tool.
Constant function(also called constant value function) means that the value does not change (i.econstant）Function of.For example, functionf（x）=4, becausefmappingAny value to 4, sofIs a constant.More generally, for a functionf: A→B, if theAAll withinxandy, bothf（x）=f（y）, so,fIs a constant function[2]。
Complex function
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definition
Complex variable function is a function whose definition field is a set of complex numbers.
The concept of complex number originates from finding the root of equation, which is in the second and third orderalgebraic equationThe square root of a negative number appears in the root of a.For a long time, peopleNumber of classesCan't understand.However, with the development of mathematics, the importance of such numbers has become increasingly apparent.The general form of complex number is a+bi, where i is the imaginary unit.
Functions with complex numbers as independent variables are called complex variable functions, and the related theory isComplex function theory。analytic functionIt is a kind of functions with analytic properties in complex variable functions. Complex variable function theory mainly studies analytic functions in the complex number field, so it is usually called complex variable function theoryAnalytic function theory[4]。
The development of complex variable function
The theory of function of complex variable came into being in the 18th century.In 1774, Euler considered in his paper thatComplex functionTwo derived by integral ofequation。Before him, French mathematiciansDarumbelIn his paper on fluid mechanics, he has already obtained them.Therefore, later people referred to these two equations and called them "the d'Alembert Euler equation".In the 19th century, the above two equations were studied in more detail when Cauchy and Riemann studied fluid mechanics, so they are also called“Cauchy Riemann condition”。
The comprehensive development of complex variable function theory was in the 19th century, just as the direct expansion of calculus ruled the mathematics of the 18th century, the new branch of complex variable function ruled the mathematics of the 19th century.At that time, mathematicians recognized that the complex variable function theory was the most abundant branch of mathematics, and called it the mathematical enjoyment of this century. Some also praised it as one of the most harmonious theories in abstract science.
Euler and d'Alembert did the earliest work for the establishment of the theory of complex variable functions. French Laplace also later studied the integration of complex variable functions, and they were all pioneers in creating this discipline.
Cauchy, Riemann and German mathematicians later laid a lot of foundations for the development of this disciplineWeierstrass。At the beginning of the 20th century, the theory of complex variable function has made great progress. The students of Weierstrass, Swedish mathematician Leffler, French mathematician Poincare, Adama, etc. have done a lot of research work, opening up a broader research field of the theory of complex variable function, and making contributions to the development of this discipline.
Complex variable function theory involves a wide range of applications, and many complex calculations are solved with it.For example, there are many different stable plane fields in physics. The so-called field is a region with physical quantities corresponding to each point. Their calculation is based onComplex functionTo solve the problem.
For example, Rukovsky of Russia used the theory of complex variable functions to solve the structural problems of aircraft wings when designing aircraft, and he also made contributions to solving problems in fluid mechanics and aviation mechanics by using the theory of complex variable functions.
The theory of complex variable function is not only widely used in other disciplines, but also in many branches of mathematics.It has gone deep into differential equation, integral equationprobability theoryAnd number theory have a great influence on their development[4]。
Contents of complex variable function
Complex variable function theory mainly includes single valued analytic function theoryRiemann surfaceTheory, geometric function theoryResidueTheory, generalized analytic function, etc.
If a function has a unique value when its variable takes a certain value, the function solution is called a single valued analytic function,polynomialThis is the function.
Complex variable functions are also studiedMultivalued functionRiemann surface theory is the main tool for studying multivalued functions.A surface composed of many layers placed together is called Riemann surface.Using this surface, we can makeSingle valueThe concepts of branch and branch point are intuitively expressed and explained in geometry.For a multivalued function, if itsRiemann surface, then, the function on the Riemannian surface becomesSingle valued function。
Riemann surface theory is a bridge between complex variable function domain and geometry, which can link the analytic properties of more profound functions with geometry.The study of Riemannian surfaces also has a relatively large impact on topology, another branch of mathematics, and gradually tends to discuss its topological properties.
In the function theory of complex variable, the content of using geometric methods to explain and solve problems is generally called geometric function theory. Complex variable function can provide geometric explanation for its properties through conformal mapping theory.derivativesEverywhere is not zeroanalytic functionThe realized images are conformal images, which are also called conformal transformations.Conformal mapping in fluid mechanicsaerodynamics、Elasticity theory、electrostatic fieldThe theory has been widely used.
ResidueTheory is an important theory in complex function theory.Residue is also calledResidueIts definition is complex.Applying residue theory toComplex functionThe calculation of integral is compared withline integralThe calculation is convenient.Calculate real variable functiondefinite integral, which can be converted into the integral of the complex variable function along the closed loop curve, and then converted into the integrand function inside the closed loop curve with the residue basic theoremIsolated singularityWhen the singular point ispoleThe calculation is more concise.
Some conditions of the single valued analytic function are appropriately changed and supplemented to meet the needs of practical research work. This changed analytic function is calledGeneralized analytic function。The generalized analytic function representsGeometryThe change of is called quasi conformal transformation.Some basic properties of analytic functions can also be applied to generalized analytic functions if they are slightly changed.
Generalized analytic functions are widely used, not only in the research of fluid mechanics, but also in solid mechanics departments such as thin shell theory.Therefore, since 2002, the theory in this field has developed very rapidly.
fromCauchyTo sum up, the theory of complex function has a history of more than 170 years.It has become an important part of mathematics with its perfect theory and exquisite skills.It has promoted the development of some disciplines and is often used as a powerful tool in practical problems. Its basic content has become a compulsory course for many majors in science and engineering.In 2002, there are still many topics to be studied in the complex variable function theory, so it will continue to develop and get more applications[4]。
Common functions
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Real function
Real function, specifying the semantic domain andrangeAll areReal number fieldFunction of.One of the characteristics of real functions is that they can draw graphs on coordinates.
Hyperbolic function
Hyperbolic sine:
Hyperbolic cosine:
Hyperbolic tangent:
Hyperbolic cotangent:
Implicit function
If the function y=f (x) of x can be determined by the equation F (x, y)=0, that is
Y is called the implicit function of x.
Where F (x, y)=0 is not a function.
Multivariate function
Multivariate function（n-Metafunction) means that the input value isn-Function of tuple.In other words, if the input range of a function isnOf setsCartesian productThis function is a subset ofn-Metafunction.
The theory of complex function, especially the analytic function, is widely used in physics.[10]Complex variable function is usually used to describe the transfer function of control system model, which is the mathematical basis of control engineering.MATLAB supports the use of complex numbers or complex matrices in functions, as well as complex variable function operations.[11]
computer
Function programming can wrap a series of statements in a function. When a piece of code needs to be used many times, you can directly write the function name to call it, so as to save code. At the same time, the function name can reflect the general function of the function, which is convenient for code reading.Functions provide parameters and return values, so that information exchange between functions can be realized, so that the functions of the whole program can be completed in collaboration.Through the function encapsulation, many complex operations are hidden inside the function. You don't need to know how a function is implemented inside, just know what the function can do, so you can better focus on the main logic of the program.[5]
Biomedicine
Logistic function is a common S-shaped function, which is widely used in biology and medicine. The feature of the function graph is that it starts to grow slowly, and then grows rapidly in a certain range. When it reaches a certain limit, the growth slows down.[6]
economics
The utility function in economics studies the goal and pursuit of people, and the meaning of utility is happiness.The essence of the so-called cost and resources in economics is personal time, because in the initial stage, individuals have not accumulated wealth and money, and what they have is time. They need to accumulate other elements through time, and the price paid is the pain and hardship of acquiring these elements and achieving these goals.[7]
Philosophy
Causality: The combination nature of scientific events is not the so-called relationship between cause and result, but the form of mathematical function.Therefore, a distinction between regular events and accidental events is to describe the combination relationship between the magnitude of these events and the related magnitude, whether it can be explained in the form of function, and whether it conforms to the concept of mathematical function.[8]
statistics
The statistical function is mainly used for various classified statistics and analysis of data areas.Statistical functions include many functions in the field of statistics, including mean function, beta distribution function, probability function, cell number calculation function, exponential and logarithmic function, maximum and minimum function, standard deviation function, variance function, normal cumulative distribution function, correlation function of data set, Pearson product moment function, t distribution function, etc.This chapter will introduce the basic syntax, parameter usage and practical application of related statistical functions through examples.[9]
