Complex variable function is a function defined in complex number field, which includes two variables: real part and imaginary part. It is widely used and of great significance in the fields of mathematics, physics and engineering.
1. Complex numbers and complex planes
The complex variable function is based on complex numbers and consists of real and imaginary parts in the form of z=x+yi, where x and y are real numbers respectively and I is imaginary unit. Complex plane represents complex numbers as points on the plane, and the real axis and imaginary axis correspond to X axis and Y axis respectively, which makes the operation of complex numbers have geometric significance.
2. Definition and properties of complex variable function
A complex variable function is a function that maps a complex domain to a complex domain, which can be expressed as w=f(z), where w and z are both complex numbers. Complex functions have many properties, including continuity, derivability and analyticity. Analytic function refers to a function that can be derived everywhere in its domain.
3. Poincare-Riemann Theorem
Poincare-Riemann theorem is an important result in the theory of complex variable functions, which establishes the relationship between analytic function and its derivative function. According to this theorem, if a function is analyzed in a certain region, no matter which direction it takes its derivative in that region, the result is the same.
4. Series representation of complex variable function
Complex function can be expressed by series expansion, the most common ones are power series and Fourier series. The power series is expanded as f(z)=∑(n = 0 to ∞) an (z-z0) n, where an is a complex coefficient. Fourier series represents a function as a series of linear combinations of sine and cosine functions.
5. Application of complex variable function
Complex function is widely used in physics, engineering and computer graphics. In physics, it can be used to describe electromagnetic fields, fluid mechanics and other problems, in engineering, it can be used to design signal processing and control systems, and in computer graphics, it can be used to generate curves and color models.
6. Relationship between Yoke Function and Analytic Function
For analytic function, its yoke function is also analytic function. * * * Yoke function is obtained by taking a negative number from the real part of a complex variable function, and its representation on the complex plane is that the function is symmetric about the real axis. Some properties and theorems can be derived by using * * * yoke function, such as * * * yoke theorem and * * * yoke relation.
7. Singularity of complex variable function
Singularity is the point of complex variable function that cannot be analyzed, including removable singularity, pole and essential singularity. Removable singularity means that the function can be analyzed by modifying the function near this point, pole means that the function diverges at this point but has good analytical properties, and essential singularity means that the function diverges at this point with complex properties.
8. Application example
Complex function is widely used, for example, in circuit analysis, the calculation can be simplified by using the technique of complex function, and the electrical properties of capacitance, inductance and resistance can be described by using the concept of complex impedance. In addition, in signal processing, complex function is often used in frequency domain analysis and filter design.
9. Summary
Complex function is an important branch of mathematics, which not only has important theoretical significance, but also has wide application value. From the geometric meaning of complex plane to the nature and application of analytic function, the learning of complex variable function involves knowledge and skills in many fields. Knowing and understanding the basic concepts and properties of complex variable function is of great significance for further study and research in related fields.