Modes and Arguments

single value function

multivalued function

Area

interior point

It is composed entirely of interior points, and any two points can be connected by a straight line.

The existence of limit must be unique

If f is differentiable at point z and its neighborhood, it is analytic at point z. If it is not analytic, it is a singular point.

If f is differentiable everywhere in the region, then f is analytic in the region, or f is an analytic function in the region.

Polynomial functions are analyzed in the complex plane

Rational functions are analyzed in the complex plane except when the denominator is zero.

no longer less than 1

is a multivalued function

If f is an analytic function in the region, and the derivative is not zero, then it constitutes a one-to-one corresponding transformation in the region, and this transformation is called a single-leaf transformation in the domain.

The analytic function implements conformal transformation at each point where the derivative is not zero.

Laplace's equation remains Poisson's equation in the new coordinate plane w under conformal transformation

zero point

an nth order pole

Expand the function into a Laurent series in the decentered neighborhood of the singular point, and take the coefficient of the negative first power (the inverse sign is required for the infinite point)

When the singular point is the extreme

There are no singular points on the real axis

The upper half plane is analytic everywhere except for a finite number of singular points.

f has no singular points on the real axis

The upper half plane is analytic everywhere except for a finite number of singular points.

the upper half-plane including the real axis,

Cauchy convergence criterion

Absolutely convergent series can arbitrarily exchange the order of the terms, and the resulting series will still converge absolutely and its sum will remain unchanged.

Two series that appear to be converging can be multiplied term by term and the series will still converge absolutely.

Ratio Discriminant Method (D'Alembert)

Radical discrimination method (Cauchy)

Gaussian discriminant

The series uniformly converges to F(z) in the region

uniform convergence

Abel's theorem

a is the singular point closest to f from the expansion center point b, then R=|a-b|

Negative terms to powers are the main part of Laurent series

Generally, Laurent expansion is obtained using known formulas

isolated singularity

non-isolated singularity

Can go to the singularity

m-order pole

pole

Singularity of nature

f(z) is continuous on l, the curve l is piecewise smooth, the complex integral exists and the above formula

Same points

Contains boundary conditions

Can be generalized to complex connected areas

Analytical function derivatives of any order

Cauchy-type integrals and integrals with parameters

Cauchy's inequality

Liouville's theorem

Module theorem

mean value theorem

Morena's theorem