In mathematics, continuity is an attribute of a function. Intuitively, a continuous function is a function in which the change of the input value is small enough and the change of the output is small enough. If a small change in the input value will cause a sudden jump, or even the output value is uncertain, the function is called a discontinuous function (or discontinuous function).
Only one-to-one functions have inverse functions. However, if a function is monotonically increasing or decreasing in its domain D, its inverse function is monotonically increasing or decreasing in its domain W ... There is symmetry between the original function and the inverse function about y = x.
Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.
Functions with complex numbers as independent variables are called complex variable functions, and the related theory is complex variable function theory. Analytic function is a kind of analytic function in complex variable function. Complex variable function theory mainly studies the analytic function in complex number field, so it is usually called analytic function theory.
The theory of complex variable function is widely used not only in other disciplines, but also in many branches of mathematics. It has been deeply involved in differential equation, integral equation, probability theory and number theory, and has had a great influence on their development.
If a function has a unique fixed value when its variable takes a certain value, then the solution of the function is called a single-valued analytic function, and a polynomial is such a function.
Complex functions also study multivalued functions, and surface theory is the main tool to study multivalued functions. A surface composed of many layers put together is called a surface. Using this surface, the concepts of single-valued bifurcation and multi-valued function bifurcation can be expressed and explained intuitively by geometry. For a multivalued function, if its surface can be made, then the function becomes a single-valued function on the surface.
In the theory of complex variable function, the content of explaining and solving problems by geometric methods is generally called geometric function theory, and complex variable function can provide geometric explanation for its properties through * * * shape mapping theory. The images realized by analytic functions whose derivatives are not zero everywhere are all * * * images, also called conformal mapping.
Residue theory is an important theory in complex variable function theory. Its definition is more complicated. It is more convenient to calculate the integral of complex variable function by residue theory than by line integral. The calculation of definite integral of real variable function can be transformed into the integral of complex variable function along closed-loop curve, and then transformed into the calculation of residue of integrand function on isolated singularity inside closed-loop curve by using the basic theorem of residue. When the singularity is the pole, the calculation is more concise.