Try to derive the Cauchy Riemann equation in the polar coordinate system

Cauchy. The Riemann equation is derived as follows: it consists of two equations: (1a) and (1b), which are mainly based on the functions of u(x,y) and v(x,y).

In general, you and v are taken as the real and imaginary parts of a complex function.

f(x + iy) = u(x,y) +iv(x,y)。If you and v are continuous on open set c, then f=u+iv is pure.

This system of equations first appeared in D'Alembert.

(d'alembert 1752)。Later Euler.

Combine this system of equations with the analytic function.

Linked (Euler 1777). Cauchy (Cauchy 1814) then employed these equations to construct his theory of functions.

The Cauchy Riemann equation is as follows:

The Cauchy Riemann equation is a mathematical tool to describe the analytic properties of complex variable functions in the complex plane. It was independently proposed by the French mathematician Cauchy and the German mathematician Riemann in the mid-19th century, respectively, hence the name Cauchy-Riemann equation. The following will explain the meaning of this equation from multiple perspectives.

1.Complex variable functions.

A complex function means that the input is a complex number, and the output is also a complex number, such as f(z)=z. Unlike real functions, in the complex plane, complex numbers can be seen as both a point and a vector with magnitude and direction, which makes the properties of complex functions more rich and complex.

2.Parsing functions.

An analytic function is a function that is derivative everywhere within its defined domain, and the derivative is also an analytic function within that defined domain. For example, e z, sinz, cosz, etc., are all analytic functions, and things like |z|, argz, etc. are not.

3.Form of Cauchy Riemann equations.

The form of the Cauchy Riemann equation is: u = v y, u y = -v x, where u(x,y) and v(x,y) are the real and imaginary parts of the complex function f(z)=u(x,y)+iv(x,y), and x and y are free variables on the complex plane.

4.The meaning of the Cauchy Riemann equation.

The form of the Cauchy Riemann equation reflects the analytic nature of the complex variable function f(z), i.e., small local changes in the complex number field can be highly ** and differentiable. The Cauchy Riemann equation is essential for the research and application in the field of complex analysis, such as the Riemann mapping theorem, conformal geometry, metapure functions, etc.

5.Applications of the Cauchy Riemann equation.

The Cauchy Riemann equation not only has a wide range of applications in complex analysis, but also plays an important role in the fields of physics, engineering, computer science, and finance. For example, in power engineering, the Cauchy Riemann equation can be used to study the distribution of current and electric potential; In finance, it is used to study volatility and the establishment of financial market models.

6.Research progress on Cauchy Riemann equations.

In recent years, with the in-depth development of mathematics and the continuous progress of computer technology, the research field of Cauchy Riemann equation has also been expanding. Emerging technologies such as complex function approximation and Gaussian process regression based on deep learning provide more advanced and efficient numerical methods for the study of Cauchy Riemann equations.

The Cauchy-Riemann equation is a sufficient and necessary condition for the complex differentiability (or total purity) of a function (Ahlfors 1953, to be precise, let f(z) = u(z) +iv(z) be a function of the complex number z c, then the complex derivative of f at the point z0 is defined if this limit exists. If this limit exists, the limit of h 0 along the real or imaginary axis can be taken; It should give the same result in both cases.

Approximation from the real axis yields the same derivative along both axes as f approximation from the imaginary axis, i.e., this is the Cauchy-Riemann equation at point z0 (2). Conversely, if f:c c is differentiable as a function mapped to r2, then f is complex differentiable if and only if the Cauchy-Riemann equation holds.

Physical Interpretation An explanation of the Cauchy-Riemann equation (pólya & szeg 0 2 1978) is not related to complex degeneration theory. Let you and v satisfy the Cauchy-Riemann equations on the open subset of r2, considering the vector field as vectors of the (real) two components. Then the second Cauchy-Riemann equation (1b) asserts that there is no rotation:

The first Cauchy-Riemann equation (1a) asserts that the vector field is passive (or scattered): according to Green's theorem and divergence theorem, respectively, such a field is conservative and has no source, with a net flow of zero over the entire open domain. (These two points are combined in the Cauchy integral theorem as real and imaginary parts.)

In fluid mechanics, such a field is a potential flow (Chanson 2000). In magnetostatics, such a vector field is a model of a magnetostatic field in a planar region without an electric current. In electrostatics, they provide a model of the electric field in a planar region that does not contain an electric charge.

In Cartesian coordinates f(z) is denoted as f(z)=u(x,y)+iv(x,y), where z is denoted as z=x+iy, similarly in polar coordinates, the variables are r and , so f(z) is denoted as f(z)=u(r, )iv(r, where z is denoted as z=re (i). Consider r and v here as intermediate variables, i.e., you and v are both composite functions about x and y, and according to the transformation relationship between polar coordinates and rectangular coordinates r= (x 2+y 2), =arctan(y x), there is u'x=u'r*r'x+u'θ*θ'x=cosθ*u'r-rsinθ*u', in the same way, find u'y,v'x and v'y, Cauchy Riemann's equation u with Cartesian coordinates'x=v'y，u'y=-v'x, get sin *u'r+rcosθ*u'θ=-cosθ*v'r+rsinθ*v'θ，cosθ*u'r-rsinθ*u'θ=sinθ*v'r+rcosθ*v', the two types can be combined to get u'r=rv'θ，v'r=-ru', which is the Cauchy Riemann equation in polar coordinates.

The Cauchy-Riemann differential square in the complex analysis is two partial differential equations that provide sufficient and necessary conditions for differentiable functions to be fully pure functions in the open set, and are named after Cauchy and Riemann Yulee. This system of equations first appeared in d'Alembert's writings. Later Euler linked this system of equations to analytic functions.

Cauchy then used these equations to construct his theory of functions. Riemann's theory of this function was published in 1851.