1. system analysis
In system analysis, Laplace transform is often used to transform the system from time domain to frequency domain. Therefore, the poles and zeros of the system can be analyzed on the complex plane. The root locus method, Nyquist diagram method and Nicholstu method for analyzing the stability of the system are all carried out on the complex plane.
Whether the poles and zeros of the system are in the left half plane or the right half plane, the root locus method is very important. If the system pole
Located in the right half plane, the causal system is unstable; If they all lie in the left half plane, the causal system is stable; On the imaginary axis, the system is critically stable. If all zeros and poles of the system are in the left half plane, it is the minimum phase system. A system is all-pass if its poles and zeros are symmetrical about the imaginary axis.
2. Signal analysis
Complex numbers used in signal analysis and other fields can conveniently represent periodic signals. The modulus value |z| represents the amplitude of the signal, and the angle arg(z) represents the phase of the sine wave at a given frequency.
Through Fourier transform, the real signal can be expressed as the sum of a series of periodic functions. These periodic functions are usually represented by the real part of a complex function of the following form:
Where ω corresponds to the angular frequency, and the complex number z contains amplitude and phase information.
In circuit analysis, the relationship between voltage and current can be easily expressed and solved by introducing the imaginary part of capacitance and inductance related to frequency. Sometimes the letter j is used as an imaginary unit to avoid confusion with the current symbol I. )
3. Generalized integral
At the application level, complex analysis is often used to calculate some real-valued abnormal functions, which are obtained from complex-valued functions. There are many methods, see contour integral method.
4. Quantum mechanics
Complex number is very important in quantum mechanics, because its theory is based on infinite Hilbert space in complex number field.
5. Relativity theory
If time variables are regarded as imaginary numbers, some space-time metric equations in special sense and general relativity can be simplified.
6. Applied Mathematics
In practical application, to solve a system with a given difference equation model, all complex characteristic roots R of the characteristic equation corresponding to the linear difference equation are usually found first, and then the system is represented by a linear combination of basis functions in the form of f (t) = e.
7. Fluid mechanics
The 2D potential flow in fluid mechanics can be described by complex variable function.
8. Split
Some fractals, such as Mandelbrot set and Julia set, are based on points on the complex plane.