It is used in many aspects.

System analysis

In system analysis, the system is often transformed from the time domain to the frequency domain through the Laplace transform. Therefore the poles and zeros of the system can be analyzed in the complex plane. The root locus method, Nyquist plot method and Nichols plot method that analyze the stability of the system are all performed on the complex plane.

The root locus method is important whether the poles and zeros of the system are in the left half plane or the right half plane. If the system pole

is located on the right half-plane, the causal system is unstable; if both are located on the left half-plane, the causal system is stable; if they are located on the imaginary axis, the system is critically stable. If all the zeros and poles of the system are in the left half plane, it is a minimum phase system. If the poles and zeros of the system are symmetric about the imaginary axis, then it is an all-pass system.

Signal Analysis

Signal analysis and other fields use complex numbers to conveniently represent periodic signals. The modulus |z| represents the amplitude of the signal, and the argument arg(z) represents the phase of the sine wave at a given frequency.

Using Fourier transform, real signals 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 form:

where ω corresponds to the angular frequency, and the complex number z contains amplitude and phase information.

In circuit analysis, introducing the imaginary parts related to capacitance, inductance and frequency can conveniently express and solve the relationship between voltage and current with simple linear equations. (The letter j is sometimes used as the imaginary unit to avoid confusion with the current symbol i.)

Abnormal integral

At the application level, complex analysis is often used to calculate certain real-valued anomalous functions. Derived from complex-valued functions. There are many methods, see the circumferential integration method.

Quantum Mechanics

Complex numbers are very important in quantum mechanics because the theory is based on the infinite-dimensional Hilbert space in the complex number field.

Relativity

If time variables are treated as imaginary numbers, some space-time metric equations in special and general relativity can be simplified.

Applied Mathematics

In practical applications, to solve a system with a given differential equation model, we usually first find out all the complex eigenvalues ??r of the characteristic equation corresponding to the linear differential equation, and then add the system Expressed as a linear combination of basis functions of the form f(t) =e.

Fluid Mechanics

Complex functions can describe two-dimensional potential flow (2D Potential Flow) in fluid mechanics.

Fractals

Some fractals such as the Mandelbrot set and the Julia set are based on points on the complex plane.

Riemann Hypothesis Trajectory

1. Decomposing prime number source numbers [development]: function []18rr+1]

1, r*6

2, 18rr--r*6+1=0

Two, the first part of plastic surgery

1, [r1+r2]*6*1/2=1

2, 18*[r1]*[r2]-[r1+r2]*6+1*1/2=0

Three, Riemann’s hypothesis turns into [prime number Distribution sphere mode]