MindMap Gallery Complex function
This is a mind map about complex functions, which summarizes analytic functions, analytic continuation, residue theory, complex function series, analytic function integrals, etc.
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The ice hockey schedule for the Milano Cortina 2026 Winter Olympics, featuring preliminary rounds, quarterfinals, and medal matches for both men's and women's tournaments from February 5–22. All game times are listed in Eastern Standard Time (EST).
This Valentine's Day brand marketing handbook provides businesses with five practical models, covering everything from creating offline experiences to driving online engagement. Whether you're a shopping mall, restaurant, or online brand, you'll find a suitable strategy: each model includes clear objectives and industry-specific guidelines, helping brands transform traffic into real sales and lasting emotional connections during this romantic season.
This Valentine's Day map illustrates love through 30 romantic possibilities, from the vintage charm of "handwritten love letters" to the urban landscape of "rooftop sunsets," from the tactile experience of a "pottery workshop" to the leisurely moments of "wine tasting at a vineyard"—offering a unique sense of occasion for every couple. Whether it's cozy, experiential, or luxurious, love always finds the most fitting expression. May you all find the perfect atmosphere for your love story.
The ice hockey schedule for the Milano Cortina 2026 Winter Olympics, featuring preliminary rounds, quarterfinals, and medal matches for both men's and women's tournaments from February 5–22. All game times are listed in Eastern Standard Time (EST).
Complex function
Analytic Functions
Complex numbers and their operations
conjugate complex numbers
Modes and Arguments
Apospheric or Riemannian sphere
Complex number arithmetic rules
Complex function
single value function
multivalued function
area
Area
interior point
It is composed entirely of interior points, and any two points can be connected by a straight line.
limits and continuity
The existence of limit must be unique
Micro-business and analytic functions
Micro Business and Differentiation
Cauchy Riemann condition
is a necessary condition for derivability
The sufficient condition must also satisfy that u and v have continuous first-order partial derivatives at point z.
Analytical functions and physical explanations
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.
harmonic function
elementary analytic functions
Power function
Polynomial functions are analyzed in the complex plane
Rational functions are analyzed in the complex plane except when the denominator is zero.
exponential function
Trigonometric functions
no longer less than 1
Analyze in the complex plane
hyperbolic function
root value function
is a multivalued function
Logarithmic function
general power function
general exponential function
Analytical function geometric properties
Single leaf transformation theorem
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.
Geometric meaning of derivatives
conformal transformation
The analytic function implements conformal transformation at each point where the derivative is not zero.
Conformal transformation properties
Laplace's equation remains Poisson's equation in the new coordinate plane w under conformal transformation
Analytical continuation
Analytical continuation
Analytical continuation
zero point
f(z) is always zero in a certain neighborhood of z0
There is a certain neighborhood of z0, in which z0 is the only zero point of f(z)
Analyze the uniqueness of continuation
gamma function
Euler integral of the second kind
beta function
Euler integrals of the first kind
Residue theory
Residue theorem
Residue theorem
an nth order pole
Residue of point at infinity
The sum of the residues in the entire plane is zero
Residue calculation method
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
Computing real integrals using residue theory
infinite integral
There are no singular points on the real axis
The upper half plane is analytic everywhere except for a finite number of singular points.
Integral involving trigonometric functions
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,
Integrals of rational expressions of trigonometric functions
Several integrals in physics problems
Dirichlet integral
Fresnel integral
Integral in heat conduction
Integrals of multivalued functions
Euler integral
Including logarithmic integral
Complex variable function series
Complex series
Complex term series
Then there is series convergence and F, F is the sum of the series
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
Complex variable function term series
The series uniformly converges to F(z) in the region
uniform convergence
continuity
term-by-term integrability
Term-by-term differentiability (Weierstrass theorem)
M discrimination method
power series
Convergence of power series
Abel's theorem
Absolute convergence within
Convergence circle and convergence radius
nature
Taylor series
Taylor's theorem
convergence range
a is the singular point closest to f from the expansion center point b, then R=|a-b|
Expand method
Laurent series
Laurent series convergence theorem
Negative terms to powers are the main part of Laurent series
Laurent's theorem
convergence range
Expand method
Generally, Laurent expansion is obtained using known formulas
Isolated Singularity of Single Value Function
Singularity of function
isolated singularity
non-isolated singularity
No matter how small the neighborhood of z=b is, there are always other singular points.
Classification of isolated singularities
Can go to the singularity
f(z) has no principal part at point b
Limits exist and are finite
f(z) is bounded in a certain decentered neighborhood of b
m-order pole
Main part limited
pole
The limit is infinity
Singularity of nature
The main part has infinite terms with negative powers
The limit does not exist
Properties of points at infinity
Can go to the singularity
Laurent expansion does not contain positive powers
m-order pole
The Laurent expansion of a point at infinity has finite terms with positive powers
Singularity of nature
There are infinite positive powers
Analytical function integral
Integral of functions of complex variables
Definition of integral of complex function
The existence conditions and calculation methods of complex integrals
f(z) is continuous on l, the curve l is piecewise smooth, the complex integral exists and the above formula
Properties of complex integrals
Same points
Cauchy's theorem
Cauchy's Theorem for Simply Connected Regions
indefinite integral
Generalization of Cauchy's theorem
Contains boundary conditions
Cauchy's theorem for complex connected regions
Cauchy's integral formula
Cauchy's formula
Can be generalized to complex connected areas
Cauchy's formula for unbounded regions
Several corollaries of Cauchy's formula
Analytical function derivatives of any order
Analyzed within the region, any order can be differentiated
Cauchy-type integrals and integrals with parameters
Cauchy's inequality
Liouville's theorem
Module theorem
|f(z)| can only obtain the maximum value on the boundary l
mean value theorem
Morena's theorem
f is continuous in the region, and the integral of any surrounding line inside it is zero, then f is analytical in the region